Step | Hyp | Ref
| Expression |
1 | | retop 22375 |
. . . 4
⊢
(topGen‘ran (,)) ∈ Top |
2 | | 0cld 20652 |
. . . 4
⊢
((topGen‘ran (,)) ∈ Top → ∅ ∈
(Clsd‘(topGen‘ran (,)))) |
3 | 1, 2 | ax-mp 5 |
. . 3
⊢ ∅
∈ (Clsd‘(topGen‘ran (,))) |
4 | | simpl3 1059 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 < (vol*‘𝐴)) |
5 | | fveq2 6103 |
. . . . . 6
⊢ (𝐴 = ∅ →
(vol*‘𝐴) =
(vol*‘∅)) |
6 | 5 | adantl 481 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) →
(vol*‘𝐴) =
(vol*‘∅)) |
7 | 4, 6 | breqtrd 4609 |
. . . 4
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → 𝑀 <
(vol*‘∅)) |
8 | | 0ss 3924 |
. . . 4
⊢ ∅
⊆ 𝐴 |
9 | 7, 8 | jctil 558 |
. . 3
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → (∅
⊆ 𝐴 ∧ 𝑀 <
(vol*‘∅))) |
10 | | sseq1 3589 |
. . . . 5
⊢ (𝑠 = ∅ → (𝑠 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴)) |
11 | | fveq2 6103 |
. . . . . 6
⊢ (𝑠 = ∅ →
(vol*‘𝑠) =
(vol*‘∅)) |
12 | 11 | breq2d 4595 |
. . . . 5
⊢ (𝑠 = ∅ → (𝑀 < (vol*‘𝑠) ↔ 𝑀 <
(vol*‘∅))) |
13 | 10, 12 | anbi12d 743 |
. . . 4
⊢ (𝑠 = ∅ → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∅ ⊆ 𝐴 ∧ 𝑀 <
(vol*‘∅)))) |
14 | 13 | rspcev 3282 |
. . 3
⊢ ((∅
∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝐴 ∧ 𝑀 < (vol*‘∅))) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
15 | 3, 9, 14 | sylancr 694 |
. 2
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 = ∅) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
16 | | mblfinlem1 32616 |
. . . 4
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
17 | 16 | 3ad2antl1 1216 |
. . 3
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) →
∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
18 | | simpl3 1059 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < (vol*‘𝐴)) |
19 | | f1ofo 6057 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
20 | | rnco2 5559 |
. . . . . . . . . . . . . . . . 17
⊢ ran ([,]
∘ 𝑓) = ([,] “
ran 𝑓) |
21 | | forn 6031 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran 𝑓 = {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
22 | 21 | imaeq2d 5385 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] “ ran 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
23 | 20, 22 | syl5eq 2656 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ 𝑓) = ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
24 | 23 | unieqd 4382 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
25 | 19, 24 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
26 | 25 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
27 | | oveq1 6556 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑢 → (𝑥 / (2↑𝑦)) = (𝑢 / (2↑𝑦))) |
28 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑢 → (𝑥 + 1) = (𝑢 + 1)) |
29 | 28 | oveq1d 6564 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑢 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑦))) |
30 | 27, 29 | opeq12d 4348 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑢 → 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉) |
31 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = 𝑣 → (2↑𝑦) = (2↑𝑣)) |
32 | 31 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑣 → (𝑢 / (2↑𝑦)) = (𝑢 / (2↑𝑣))) |
33 | 31 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑣 → ((𝑢 + 1) / (2↑𝑦)) = ((𝑢 + 1) / (2↑𝑣))) |
34 | 32, 33 | opeq12d 4348 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑣 → 〈(𝑢 / (2↑𝑦)), ((𝑢 + 1) / (2↑𝑦))〉 = 〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
35 | 30, 34 | cbvmpt2v 6633 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑢 ∈ ℤ, 𝑣 ∈ ℕ0 ↦
〈(𝑢 / (2↑𝑣)), ((𝑢 + 1) / (2↑𝑣))〉) |
36 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑧 → ([,]‘𝑎) = ([,]‘𝑧)) |
37 | 36 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑧 → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘𝑧) ⊆ ([,]‘𝑐))) |
38 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑧 → (𝑎 = 𝑐 ↔ 𝑧 = 𝑐)) |
39 | 37, 38 | imbi12d 333 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑧 → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
40 | 39 | ralbidv 2969 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑧 → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐))) |
41 | 40 | cbvrabv 3172 |
. . . . . . . . . . . . . . 15
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} = {𝑧 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑧) ⊆ ([,]‘𝑐) → 𝑧 = 𝑐)} |
42 | | ssrab2 3650 |
. . . . . . . . . . . . . . . 16
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
{𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
44 | 35, 41, 43 | dyadmbllem 23173 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,]
“ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
45 | 44 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,]
“ {𝑏 ∈ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴}) = ∪ ([,] “ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})) |
46 | 26, 45 | eqtr4d 2647 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴})) |
47 | | opnmbllem0 32615 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
48 | 47 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∪ ([,] “ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
49 | 48 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ([,]
“ {𝑏 ∈ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣
([,]‘𝑏) ⊆ 𝐴}) = 𝐴) |
50 | 46, 49 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∪ ran
([,] ∘ 𝑓) = 𝐴) |
51 | 50 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘∪ ran ([,] ∘ 𝑓)) = (vol*‘𝐴)) |
52 | | f1of 6050 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
53 | | ssrab2 3650 |
. . . . . . . . . . . . . 14
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} |
54 | 35 | dyadf 23165 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶( ≤ ∩ (ℝ ×
ℝ)) |
55 | | frn 5966 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶( ≤ ∩ (ℝ × ℝ)) → ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ ( ≤ ∩
(ℝ × ℝ))) |
56 | 54, 55 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ ( ≤ ∩
(ℝ × ℝ)) |
57 | 42, 56 | sstri 3577 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ ( ≤ ∩ (ℝ ×
ℝ)) |
58 | 53, 57 | sstri 3577 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ ×
ℝ)) |
59 | | fss 5969 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ( ≤ ∩ (ℝ ×
ℝ))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
60 | 52, 58, 59 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
61 | 53, 42 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
62 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
63 | 61, 62 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
64 | 63 | adantrr 749 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
65 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
66 | 61, 65 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
67 | 66 | adantrl 748 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) |
68 | 35 | dyaddisj 23170 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓‘𝑚) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∧ (𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
69 | 64, 67, 68 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
70 | 52, 69 | sylan 487 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
71 | | df-3or 1032 |
. . . . . . . . . . . . . . . 16
⊢
((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
72 | 70, 71 | sylib 207 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
73 | | elrabi 3328 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) |
74 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑚) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑚))) |
75 | 74 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑚) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐))) |
76 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑚) → (𝑎 = 𝑐 ↔ (𝑓‘𝑚) = 𝑐)) |
77 | 75, 76 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = (𝑓‘𝑚) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
78 | 77 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎 = (𝑓‘𝑚) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
79 | 78 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐))) |
80 | 79 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)) |
81 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑧) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑧))) |
82 | 81 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑧) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)))) |
83 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑧) → ((𝑓‘𝑚) = 𝑐 ↔ (𝑓‘𝑚) = (𝑓‘𝑧))) |
84 | 82, 83 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = (𝑓‘𝑧) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐) ↔ (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧)))) |
85 | 84 | rspcva 3280 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘𝑐) → (𝑓‘𝑚) = 𝑐)) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
86 | 73, 80, 85 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
87 | | elrabi 3328 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴}) |
88 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑎 = (𝑓‘𝑧) → ([,]‘𝑎) = ([,]‘(𝑓‘𝑧))) |
89 | 88 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑧) → (([,]‘𝑎) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐))) |
90 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = (𝑓‘𝑧) → (𝑎 = 𝑐 ↔ (𝑓‘𝑧) = 𝑐)) |
91 | 89, 90 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = (𝑓‘𝑧) → ((([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
92 | 91 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = (𝑓‘𝑧) → (∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐) ↔ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
93 | 92 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ↔ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐))) |
94 | 93 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)) |
95 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑐 = (𝑓‘𝑚) → ([,]‘𝑐) = ([,]‘(𝑓‘𝑚))) |
96 | 95 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑚) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) ↔ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)))) |
97 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = (𝑓‘𝑚) → ((𝑓‘𝑧) = 𝑐 ↔ (𝑓‘𝑧) = (𝑓‘𝑚))) |
98 | 96, 97 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = (𝑓‘𝑚) → ((([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐) ↔ (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚)))) |
99 | 98 | rspcva 3280 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓‘𝑚) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∧ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘𝑐) → (𝑓‘𝑧) = 𝑐)) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))) |
100 | 87, 94, 99 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑧) = (𝑓‘𝑚))) |
101 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓‘𝑧) = (𝑓‘𝑚) ↔ (𝑓‘𝑚) = (𝑓‘𝑧)) |
102 | 100, 101 | syl6ib 240 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚)) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
103 | 86, 102 | jaod 394 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓‘𝑚) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑓‘𝑧) ∈ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
104 | 62, 65, 103 | syl2an 493 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) ∧ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
105 | 104 | anandis 869 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
106 | 52, 105 | sylan 487 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → (𝑓‘𝑚) = (𝑓‘𝑧))) |
107 | | f1of1 6049 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
108 | | f1veqaeq 6418 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ–1-1→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧)) |
109 | 107, 108 | sylan 487 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((𝑓‘𝑚) = (𝑓‘𝑧) → 𝑚 = 𝑧)) |
110 | 106, 109 | syld 46 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → ((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) → 𝑚 = 𝑧)) |
111 | 110 | orim1d 880 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (((([,]‘(𝑓‘𝑚)) ⊆ ([,]‘(𝑓‘𝑧)) ∨ ([,]‘(𝑓‘𝑧)) ⊆ ([,]‘(𝑓‘𝑚))) ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅))) |
112 | 72, 111 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
113 | 112 | ralrimivva 2954 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
114 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → (𝑚 = 𝑝 ↔ 𝑧 = 𝑝)) |
115 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = 𝑧 → (𝑓‘𝑚) = (𝑓‘𝑧)) |
116 | 115 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 = 𝑧 → ((,)‘(𝑓‘𝑚)) = ((,)‘(𝑓‘𝑧))) |
117 | 116 | ineq1d 3775 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝)))) |
118 | 117 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅ ↔ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
119 | 114, 118 | orbi12d 742 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑧 → ((𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
120 | 119 | ralbidv 2969 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑧 → (∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
121 | 120 | cbvralv 3147 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
ℕ ∀𝑝 ∈
ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
122 | | eqeq2 2621 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑝 → (𝑚 = 𝑧 ↔ 𝑚 = 𝑝)) |
123 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑝 → (𝑓‘𝑧) = (𝑓‘𝑝)) |
124 | 123 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = 𝑝 → ((,)‘(𝑓‘𝑧)) = ((,)‘(𝑓‘𝑝))) |
125 | 124 | ineq2d 3776 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 𝑝 → (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝)))) |
126 | 125 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑝 → ((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
127 | 122, 126 | orbi12d 742 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑝 → ((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅))) |
128 | 127 | cbvralv 3147 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑧 ∈
ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
129 | 128 | ralbii 2963 |
. . . . . . . . . . . . . 14
⊢
(∀𝑚 ∈
ℕ ∀𝑧 ∈
ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ ∀𝑚 ∈ ℕ ∀𝑝 ∈ ℕ (𝑚 = 𝑝 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
130 | 124 | disjor 4567 |
. . . . . . . . . . . . . 14
⊢
(Disj 𝑧
∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑧 ∈ ℕ ∀𝑝 ∈ ℕ (𝑧 = 𝑝 ∨ (((,)‘(𝑓‘𝑧)) ∩ ((,)‘(𝑓‘𝑝))) = ∅)) |
131 | 121, 129,
130 | 3bitr4ri 292 |
. . . . . . . . . . . . 13
⊢
(Disj 𝑧
∈ ℕ ((,)‘(𝑓‘𝑧)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅)) |
132 | 113, 131 | sylibr 223 |
. . . . . . . . . . . 12
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑧 ∈ ℕ ((,)‘(𝑓‘𝑧))) |
133 | | eqid 2610 |
. . . . . . . . . . . 12
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
134 | 60, 132, 133 | uniiccvol 23154 |
. . . . . . . . . . 11
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
135 | 134 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘∪ ran ([,] ∘ 𝑓)) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
136 | 51, 135 | eqtr3d 2646 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (vol*‘𝐴) = sup(ran seq1( + , ((abs ∘ −
) ∘ 𝑓)),
ℝ*, < )) |
137 | 18, 136 | breqtrd 4609 |
. . . . . . . 8
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → 𝑀 < sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
138 | | absf 13925 |
. . . . . . . . . . . 12
⊢
abs:ℂ⟶ℝ |
139 | | subf 10162 |
. . . . . . . . . . . 12
⊢ −
:(ℂ × ℂ)⟶ℂ |
140 | | fco 5971 |
. . . . . . . . . . . 12
⊢
((abs:ℂ⟶ℝ ∧ − :(ℂ ×
ℂ)⟶ℂ) → (abs ∘ − ):(ℂ ×
ℂ)⟶ℝ) |
141 | 138, 139,
140 | mp2an 704 |
. . . . . . . . . . 11
⊢ (abs
∘ − ):(ℂ × ℂ)⟶ℝ |
142 | | zre 11258 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℤ → 𝑥 ∈
ℝ) |
143 | | 2re 10967 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℝ |
144 | | reexpcl 12739 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℝ ∧ 𝑦
∈ ℕ0) → (2↑𝑦) ∈ ℝ) |
145 | 143, 144 | mpan 702 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℝ) |
146 | | 2cn 10968 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ∈
ℂ |
147 | | 2ne0 10990 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 2 ≠
0 |
148 | | nn0z 11277 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℤ) |
149 | | expne0i 12754 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((2
∈ ℂ ∧ 2 ≠ 0 ∧ 𝑦 ∈ ℤ) → (2↑𝑦) ≠ 0) |
150 | 146, 147,
148, 149 | mp3an12i 1420 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ≠
0) |
151 | 145, 150 | jca 553 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℕ0
→ ((2↑𝑦) ∈
ℝ ∧ (2↑𝑦)
≠ 0)) |
152 | | redivcl 10623 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ (𝑥 / (2↑𝑦)) ∈
ℝ) |
153 | | peano2re 10088 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
154 | | redivcl 10623 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑥 + 1) ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
155 | 153, 154 | syl3an1 1351 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ ((𝑥 + 1) /
(2↑𝑦)) ∈
ℝ) |
156 | 152, 155 | opelxpd 5073 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ∈ ℝ ∧
(2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
157 | 156 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ℝ ∧
((2↑𝑦) ∈ ℝ
∧ (2↑𝑦) ≠ 0))
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
158 | 142, 151,
157 | syl2an 493 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ0)
→ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ)) |
159 | 158 | rgen2 2958 |
. . . . . . . . . . . . . . . . 17
⊢
∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ ×
ℝ) |
160 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) |
161 | 160 | fmpt2 7126 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
ℤ ∀𝑦 ∈
ℕ0 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉 ∈ (ℝ × ℝ)
↔ (𝑥 ∈ ℤ,
𝑦 ∈
ℕ0 ↦ 〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ)) |
162 | 159, 161 | mpbi 219 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) |
163 | | frn 5966 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉):(ℤ ×
ℕ0)⟶(ℝ × ℝ) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ)) |
164 | 162, 163 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑥 ∈ ℤ, 𝑦 ∈ ℕ0
↦ 〈(𝑥 /
(2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ⊆ (ℝ
× ℝ) |
165 | 42, 164 | sstri 3577 |
. . . . . . . . . . . . . 14
⊢ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ⊆ (ℝ ×
ℝ) |
166 | 53, 165 | sstri 3577 |
. . . . . . . . . . . . 13
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ ×
ℝ) |
167 | | ax-resscn 9872 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℂ |
168 | | xpss12 5148 |
. . . . . . . . . . . . . 14
⊢ ((ℝ
⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℝ × ℝ)
⊆ (ℂ × ℂ)) |
169 | 167, 167,
168 | mp2an 704 |
. . . . . . . . . . . . 13
⊢ (ℝ
× ℝ) ⊆ (ℂ × ℂ) |
170 | 166, 169 | sstri 3577 |
. . . . . . . . . . . 12
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ ×
ℂ) |
171 | | fss 5969 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℂ × ℂ)) →
𝑓:ℕ⟶(ℂ
× ℂ)) |
172 | 170, 171 | mpan2 703 |
. . . . . . . . . . 11
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓:ℕ⟶(ℂ ×
ℂ)) |
173 | | fco 5971 |
. . . . . . . . . . 11
⊢ (((abs
∘ − ):(ℂ × ℂ)⟶ℝ ∧ 𝑓:ℕ⟶(ℂ ×
ℂ)) → ((abs ∘ − ) ∘ 𝑓):ℕ⟶ℝ) |
174 | 141, 172,
173 | sylancr 694 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘
𝑓):ℕ⟶ℝ) |
175 | | nnuz 11599 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
176 | | 1z 11284 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℤ |
177 | 176 | a1i 11 |
. . . . . . . . . . 11
⊢ (((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ → 1 ∈
ℤ) |
178 | | ffvelrn 6265 |
. . . . . . . . . . 11
⊢ ((((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ ∧ 𝑛 ∈ ℕ) → (((abs
∘ − ) ∘ 𝑓)‘𝑛) ∈ ℝ) |
179 | 175, 177,
178 | serfre 12692 |
. . . . . . . . . 10
⊢ (((abs
∘ − ) ∘ 𝑓):ℕ⟶ℝ → seq1( + ,
((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ) |
180 | | frn 5966 |
. . . . . . . . . . 11
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( +
, ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ) |
181 | | ressxr 9962 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℝ* |
182 | 180, 181 | syl6ss 3580 |
. . . . . . . . . 10
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶ℝ → ran seq1( +
, ((abs ∘ − ) ∘ 𝑓)) ⊆
ℝ*) |
183 | 52, 174, 179, 182 | 4syl 19 |
. . . . . . . . 9
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ −
) ∘ 𝑓)) ⊆
ℝ*) |
184 | | rexr 9964 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℝ → 𝑀 ∈
ℝ*) |
185 | 184 | 3ad2ant2 1076 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
𝑀 ∈
ℝ*) |
186 | | supxrlub 12027 |
. . . . . . . . 9
⊢ ((ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* ∧ 𝑀 ∈ ℝ*)
→ (𝑀 < sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ↔
∃𝑧 ∈ ran seq1( +
, ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧)) |
187 | 183, 185,
186 | syl2anr 494 |
. . . . . . . 8
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (𝑀 < sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ↔ ∃𝑧 ∈ ran seq1( + , ((abs ∘ − )
∘ 𝑓))𝑀 < 𝑧)) |
188 | 137, 187 | mpbid 221 |
. . . . . . 7
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑧 ∈ ran seq1( + , ((abs ∘ − )
∘ 𝑓))𝑀 < 𝑧) |
189 | | seqfn 12675 |
. . . . . . . . . 10
⊢ (1 ∈
ℤ → seq1( + , ((abs ∘ − ) ∘ 𝑓)) Fn
(ℤ≥‘1)) |
190 | 176, 189 | ax-mp 5 |
. . . . . . . . 9
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) Fn
(ℤ≥‘1) |
191 | 175 | fneq2i 5900 |
. . . . . . . . 9
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)) Fn ℕ ↔ seq1( + , ((abs ∘
− ) ∘ 𝑓)) Fn
(ℤ≥‘1)) |
192 | 190, 191 | mpbir 220 |
. . . . . . . 8
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) Fn ℕ |
193 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑧 = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛) → (𝑀 < 𝑧 ↔ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛))) |
194 | 193 | rexrn 6269 |
. . . . . . . 8
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)) Fn ℕ → (∃𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛))) |
195 | 192, 194 | ax-mp 5 |
. . . . . . 7
⊢
(∃𝑧 ∈ ran
seq1( + , ((abs ∘ − ) ∘ 𝑓))𝑀 < 𝑧 ↔ ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
196 | 188, 195 | sylib 207 |
. . . . . 6
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
197 | 60 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
198 | | 0le0 10987 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ≤
0 |
199 | | df-br 4584 |
. . . . . . . . . . . . . . . . . 18
⊢ (0 ≤ 0
↔ 〈0, 0〉 ∈ ≤ ) |
200 | 198, 199 | mpbi 219 |
. . . . . . . . . . . . . . . . 17
⊢ 〈0,
0〉 ∈ ≤ |
201 | | 0re 9919 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℝ |
202 | | opelxpi 5072 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℝ ∧ 0 ∈ ℝ) → 〈0, 0〉 ∈ (ℝ
× ℝ)) |
203 | 201, 201,
202 | mp2an 704 |
. . . . . . . . . . . . . . . . 17
⊢ 〈0,
0〉 ∈ (ℝ × ℝ) |
204 | | elin 3758 |
. . . . . . . . . . . . . . . . 17
⊢ (〈0,
0〉 ∈ ( ≤ ∩ (ℝ × ℝ)) ↔ (〈0, 0〉
∈ ≤ ∧ 〈0, 0〉 ∈ (ℝ ×
ℝ))) |
205 | 200, 203,
204 | mpbir2an 957 |
. . . . . . . . . . . . . . . 16
⊢ 〈0,
0〉 ∈ ( ≤ ∩ (ℝ × ℝ)) |
206 | | ifcl 4080 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓‘𝑧) ∈ ( ≤ ∩ (ℝ ×
ℝ)) ∧ 〈0, 0〉 ∈ ( ≤ ∩ (ℝ ×
ℝ))) → if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ ( ≤ ∩
(ℝ × ℝ))) |
207 | 197, 205,
206 | sylancl 693 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ ( ≤ ∩
(ℝ × ℝ))) |
208 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) |
209 | 207, 208 | fmptd 6292 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶( ≤
∩ (ℝ × ℝ))) |
210 | | df-ov 6552 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0(,)0) =
((,)‘〈0, 0〉) |
211 | | iooid 12074 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (0(,)0) =
∅ |
212 | 210, 211 | eqtr3i 2634 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((,)‘〈0, 0〉) = ∅ |
213 | 212 | ineq1i 3772 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = (∅ ∩ ((,)‘(𝑓‘𝑧))) |
214 | | 0in 3921 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (∅
∩ ((,)‘(𝑓‘𝑧))) = ∅ |
215 | 213, 214 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = ∅ |
216 | 215 | olci 405 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅) |
217 | | ineq1 3769 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧)))) |
218 | 217 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
219 | 218 | orbi2d 734 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘(𝑓‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) =
∅))) |
220 | | ineq1 3769 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧)))) |
221 | 220 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((((,)‘〈0, 0〉) ∩ ((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
222 | 221 | orbi2d 734 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) =
∅))) |
223 | 219, 222 | ifboth 4074 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 = 𝑧 ∨ (((,)‘(𝑓‘𝑚)) ∩ ((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (((,)‘〈0, 0〉) ∩
((,)‘(𝑓‘𝑧))) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
224 | 112, 216,
223 | sylancl 693 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅)) |
225 | 212 | ineq2i 3773 |
. . . . . . . . . . . . . . . . . . 19
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
∅) |
226 | | in0 3920 |
. . . . . . . . . . . . . . . . . . 19
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
∅) = ∅ |
227 | 225, 226 | eqtri 2632 |
. . . . . . . . . . . . . . . . . 18
⊢ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅ |
228 | 227 | olci 405 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅) |
229 | | ineq2 3770 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
(if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉)))) |
230 | 229 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
231 | 230 | orbi2d 734 |
. . . . . . . . . . . . . . . . . 18
⊢
(((,)‘(𝑓‘𝑧)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅))) |
232 | | ineq2 3770 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
(if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉)))) |
233 | 232 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅ ↔ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
234 | 233 | orbi2d 734 |
. . . . . . . . . . . . . . . . . 18
⊢
(((,)‘〈0, 0〉) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉)) →
((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅) ↔ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅))) |
235 | 231, 234 | ifboth 4074 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘(𝑓‘𝑧))) = ∅) ∧ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
((,)‘〈0, 0〉)) = ∅)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
236 | 224, 228,
235 | sylancl 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑚 ∈ ℕ ∧ 𝑧 ∈ ℕ)) → (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
237 | 236 | ralrimivva 2954 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
238 | | disjeq2 4557 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑚 ∈
ℕ ((,)‘((𝑧
∈ ℕ ↦ if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) →
(Disj 𝑚 ∈
ℕ ((,)‘((𝑧
∈ ℕ ↦ if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉)))) |
239 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑚 → (𝑧 ∈ (1...𝑛) ↔ 𝑚 ∈ (1...𝑛))) |
240 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝑚 → (𝑓‘𝑧) = (𝑓‘𝑚)) |
241 | 239, 240 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝑚 → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) |
242 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓‘𝑚) ∈ V |
243 | | opex 4859 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 〈0,
0〉 ∈ V |
244 | 242, 243 | ifex 4106 |
. . . . . . . . . . . . . . . . . . . 20
⊢ if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉) ∈
V |
245 | 241, 208,
244 | fvmpt 6191 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℕ → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) |
246 | 245 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 ∈ ℕ →
((,)‘((𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((,)‘if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉))) |
247 | | fvif 6114 |
. . . . . . . . . . . . . . . . . 18
⊢
((,)‘if(𝑚
∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉)) |
248 | 246, 247 | syl6eq 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ →
((,)‘((𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉))) |
249 | 238, 248 | mprg 2910 |
. . . . . . . . . . . . . . . 16
⊢
(Disj 𝑚
∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ Disj 𝑚 ∈ ℕ if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0,
0〉))) |
250 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑧 → (𝑚 ∈ (1...𝑛) ↔ 𝑧 ∈ (1...𝑛))) |
251 | 250, 116 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑧 → if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0,
0〉))) |
252 | 251 | disjor 4567 |
. . . . . . . . . . . . . . . 16
⊢
(Disj 𝑚
∈ ℕ if(𝑚 ∈
(1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ↔
∀𝑚 ∈ ℕ
∀𝑧 ∈ ℕ
(𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
253 | 249, 252 | bitri 263 |
. . . . . . . . . . . . . . 15
⊢
(Disj 𝑚
∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) ↔ ∀𝑚 ∈ ℕ ∀𝑧 ∈ ℕ (𝑚 = 𝑧 ∨ (if(𝑚 ∈ (1...𝑛), ((,)‘(𝑓‘𝑚)), ((,)‘〈0, 0〉)) ∩
if(𝑧 ∈ (1...𝑛), ((,)‘(𝑓‘𝑧)), ((,)‘〈0, 0〉))) =
∅)) |
254 | 237, 253 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → Disj 𝑚 ∈ ℕ ((,)‘((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
255 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
256 | 209, 254,
255 | uniiccvol 23154 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< )) |
257 | 256 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) = sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< )) |
258 | | rexpssxrxp 9963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
259 | 166, 258 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . 20
⊢ {𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ⊆ (ℝ* ×
ℝ*) |
260 | 259, 65 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ* ×
ℝ*)) |
261 | | 0xr 9965 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ∈
ℝ* |
262 | | opelxpi 5072 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((0
∈ ℝ* ∧ 0 ∈ ℝ*) → 〈0,
0〉 ∈ (ℝ* ×
ℝ*)) |
263 | 261, 261,
262 | mp2an 704 |
. . . . . . . . . . . . . . . . . . 19
⊢ 〈0,
0〉 ∈ (ℝ* ×
ℝ*) |
264 | | ifcl 4080 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓‘𝑧) ∈ (ℝ* ×
ℝ*) ∧ 〈0, 0〉 ∈ (ℝ* ×
ℝ*)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
(ℝ* × ℝ*)) |
265 | 260, 263,
264 | sylancl 693 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
(ℝ* × ℝ*)) |
266 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
267 | | iccf 12143 |
. . . . . . . . . . . . . . . . . . . 20
⊢
[,]:(ℝ* × ℝ*)⟶𝒫
ℝ* |
268 | 267 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,]:(ℝ* ×
ℝ*)⟶𝒫 ℝ*) |
269 | 268 | feqmptd 6159 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → [,] = (𝑚 ∈ (ℝ* ×
ℝ*) ↦ ([,]‘𝑚))) |
270 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) → ([,]‘𝑚) = ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
271 | 265, 266,
269, 270 | fmptco 6303 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
272 | 52, 271 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
273 | 272 | rneqd 5274 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = ran (𝑧 ∈ ℕ ↦
([,]‘if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
274 | 273 | unieqd 4382 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪ ran
([,] ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) |
275 | | peano2nn 10909 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
276 | 275, 175 | syl6eleq 2698 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
(ℤ≥‘1)) |
277 | | fzouzsplit 12372 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 + 1) ∈
(ℤ≥‘1) → (ℤ≥‘1) =
((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
278 | 276, 277 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ →
(ℤ≥‘1) = ((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
279 | 175, 278 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ → ℕ =
((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
280 | | nnz 11276 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
281 | | fzval3 12404 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℤ →
(1...𝑛) = (1..^(𝑛 + 1))) |
282 | 280, 281 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ →
(1...𝑛) = (1..^(𝑛 + 1))) |
283 | 282 | uneq1d 3728 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ →
((1...𝑛) ∪
(ℤ≥‘(𝑛 + 1))) = ((1..^(𝑛 + 1)) ∪
(ℤ≥‘(𝑛 + 1)))) |
284 | 279, 283 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → ℕ =
((1...𝑛) ∪
(ℤ≥‘(𝑛 + 1)))) |
285 | | fvif 6114 |
. . . . . . . . . . . . . . . . . 18
⊢
([,]‘if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉)) |
286 | 285 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ →
([,]‘if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
287 | 284, 286 | iuneq12d 4482 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
288 | | fvex 6113 |
. . . . . . . . . . . . . . . . 17
⊢
([,]‘if(𝑧
∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) ∈
V |
289 | 288 | dfiun3 5301 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑧 ∈ ℕ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
290 | | iunxun 4541 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑧 ∈ ((1...𝑛) ∪ (ℤ≥‘(𝑛 + 1)))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉))) |
291 | 287, 289,
290 | 3eqtr3g 2667 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0,
0〉)))) |
292 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) =
([,]‘(𝑓‘𝑧))) |
293 | 292 | iuneq2i 4475 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) |
294 | 293 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))) |
295 | | uznfz 12292 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈
(ℤ≥‘(𝑛 + 1)) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1))) |
296 | 295 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...((𝑛 + 1) − 1))) |
297 | | nncn 10905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℂ) |
298 | | ax-1cn 9873 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 1 ∈
ℂ |
299 | | pncan 10166 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑛 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑛 + 1)
− 1) = 𝑛) |
300 | 297, 298,
299 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ℕ → ((𝑛 + 1) − 1) = 𝑛) |
301 | 300 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ ℕ →
(1...((𝑛 + 1) − 1)) =
(1...𝑛)) |
302 | 301 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 ∈ ℕ → (𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ 𝑧 ∈ (1...𝑛))) |
303 | 302 | notbid 307 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ ℕ → (¬
𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬
𝑧 ∈ (1...𝑛))) |
304 | 303 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → (¬ 𝑧 ∈ (1...((𝑛 + 1) − 1)) ↔ ¬ 𝑧 ∈ (1...𝑛))) |
305 | 296, 304 | mpbid 221 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → ¬ 𝑧 ∈ (1...𝑛)) |
306 | 305 | iffalsed 4047 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ ℕ ∧ 𝑧 ∈
(ℤ≥‘(𝑛 + 1))) → if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) =
([,]‘〈0, 0〉)) |
307 | 306 | iuneq2dv 4478 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) = ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) |
308 | 294, 307 | uneq12d 3730 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → (∪ 𝑧 ∈ (1...𝑛)if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉)) ∪
∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))if(𝑧 ∈ (1...𝑛), ([,]‘(𝑓‘𝑧)), ([,]‘〈0, 0〉))) =
(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
309 | 291, 308 | eqtrd 2644 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ∪ ran (𝑧 ∈ ℕ ↦ ([,]‘if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
310 | 274, 309 | sylan9eq 2664 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) = (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉))) |
311 | 310 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ ran ([,] ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) =
(vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)))) |
312 | | xrltso 11850 |
. . . . . . . . . . . . . . 15
⊢ < Or
ℝ* |
313 | 312 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → < Or
ℝ*) |
314 | | elnnuz 11600 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈
(ℤ≥‘1)) |
315 | 314 | biimpi 205 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
(ℤ≥‘1)) |
316 | 315 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → 𝑛 ∈
(ℤ≥‘1)) |
317 | | elfznn 12241 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 ∈ (1...𝑛) → 𝑢 ∈ ℕ) |
318 | 174 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑢) ∈ ℝ) |
319 | 317, 318 | sylan2 490 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑢) ∈ ℝ) |
320 | 319 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑢 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑢) ∈ ℝ) |
321 | | readdcl 9898 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 + 𝑣) ∈ ℝ) |
322 | 321 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑢 + 𝑣) ∈ ℝ) |
323 | 316, 320,
322 | seqcl 12683 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
324 | 323 | rexrd 9968 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈
ℝ*) |
325 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
326 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈ (1...𝑛) → if(𝑚 ∈ (1...𝑛), (𝑓‘𝑚), 〈0, 0〉) = (𝑓‘𝑚)) |
327 | 241, 326 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 ∈ (1...𝑛) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑚)) |
328 | | elfznn 12241 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → 𝑚 ∈ ℕ) |
329 | 242 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈ (1...𝑛) → (𝑓‘𝑚) ∈ V) |
330 | 325, 327,
328, 329 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ (1...𝑛) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
331 | 330 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
332 | 331 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘(𝑓‘𝑚))) |
333 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓‘𝑧) ∈ V |
334 | 333, 243 | ifex 4106 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈
V |
335 | 334, 208 | fnmpti 5935 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) Fn
ℕ |
336 | | fvco2 6183 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
337 | 335, 328,
336 | sylancr 694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ (1...𝑛) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
338 | 337 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
339 | | ffn 5958 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 𝑓 Fn ℕ) |
340 | | fvco2 6183 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 Fn ℕ ∧ 𝑚 ∈ ℕ) → (((abs
∘ − ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
341 | 339, 328,
340 | syl2an 493 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
342 | 332, 338,
341 | 3eqtr4d 2654 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
343 | 342 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
344 | 316, 343 | seqfveq 12687 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
345 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → 1 ∈ ℤ) |
346 | 170, 65 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℂ ×
ℂ)) |
347 | | 0cn 9911 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 0 ∈
ℂ |
348 | | opelxpi 5072 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((0
∈ ℂ ∧ 0 ∈ ℂ) → 〈0, 0〉 ∈ (ℂ
× ℂ)) |
349 | 347, 347,
348 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 〈0,
0〉 ∈ (ℂ × ℂ) |
350 | | ifcl 4080 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓‘𝑧) ∈ (ℂ × ℂ) ∧
〈0, 0〉 ∈ (ℂ × ℂ)) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ (ℂ ×
ℂ)) |
351 | 346, 349,
350 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) ∈ (ℂ ×
ℂ)) |
352 | 351, 208 | fmptd 6292 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶(ℂ
× ℂ)) |
353 | | fco 5971 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((abs
∘ − ):(ℂ × ℂ)⟶ℝ ∧ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)):ℕ⟶(ℂ
× ℂ)) → ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0,
0〉))):ℕ⟶ℝ) |
354 | 141, 352,
353 | sylancr 694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0,
0〉))):ℕ⟶ℝ) |
355 | 354 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) ∈
ℝ) |
356 | 175, 345,
355 | serfre 12692 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0,
0〉)))):ℕ⟶ℝ) |
357 | 356 | ffnd 5959 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn
ℕ) |
358 | | fnfvelrn 6264 |
. . . . . . . . . . . . . . . 16
⊢ ((seq1( +
, ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn ℕ ∧
𝑛 ∈ ℕ) →
(seq1( + , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
359 | 357, 358 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
360 | 344, 359 | eqeltrrd 2689 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) |
361 | | frn 5966 |
. . . . . . . . . . . . . . . . . 18
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0,
0〉)))):ℕ⟶ℝ → ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) ⊆
ℝ) |
362 | 356, 361 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ran seq1( + , ((abs ∘ −
) ∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) ⊆
ℝ) |
363 | 362 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) ⊆
ℝ) |
364 | 363 | sselda 3568 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → 𝑚 ∈
ℝ) |
365 | 323 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → (seq1( + ,
((abs ∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
366 | | readdcl 9898 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑚 + 𝑢) ∈ ℝ) |
367 | 366 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ)) → (𝑚 + 𝑢) ∈ ℝ) |
368 | | recn 9905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑚 ∈ ℝ → 𝑚 ∈
ℂ) |
369 | | recn 9905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑢 ∈ ℝ → 𝑢 ∈
ℂ) |
370 | | recn 9905 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑣 ∈ ℝ → 𝑣 ∈
ℂ) |
371 | | addass 9902 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
372 | 368, 369,
370, 371 | syl3an 1360 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
373 | 372 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ (𝑚 ∈ ℝ ∧ 𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → ((𝑚 + 𝑢) + 𝑣) = (𝑚 + (𝑢 + 𝑣))) |
374 | | nnltp1le 11310 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 < 𝑡 ↔ (𝑛 + 1) ≤ 𝑡)) |
375 | 374 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑛 + 1) ≤ 𝑡) |
376 | 275 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℤ) |
377 | | nnz 11276 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
ℤ) |
378 | | eluz 11577 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑛 + 1) ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
379 | 376, 377,
378 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
380 | 379 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (𝑡 ∈ (ℤ≥‘(𝑛 + 1)) ↔ (𝑛 + 1) ≤ 𝑡)) |
381 | 375, 380 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1))) |
382 | 381 | adantlll 750 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑡 ∈ (ℤ≥‘(𝑛 + 1))) |
383 | 315 | ad3antlr 763 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑛 ∈
(ℤ≥‘1)) |
384 | | simplll 794 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
385 | | elfznn 12241 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℕ) |
386 | 384, 385,
355 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) ∈
ℝ) |
387 | 367, 373,
382, 383, 386 | seqsplit 12696 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = ((seq1( + , ((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡))) |
388 | 344 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
389 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℤ) |
390 | 389 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ) |
391 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ) |
392 | 275 | nnred 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ) |
393 | 392 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ) |
394 | 389 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ ℝ) |
395 | 394 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ) |
396 | 275 | nngt0d 10941 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑛 ∈ ℕ → 0 <
(𝑛 + 1)) |
397 | 396 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1)) |
398 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → (𝑛 + 1) ≤ 𝑚) |
399 | 398 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚) |
400 | 391, 393,
395, 397, 399 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚) |
401 | | elnnz 11264 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℕ ↔ (𝑚 ∈ ℤ ∧ 0 <
𝑚)) |
402 | 390, 400,
401 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ) |
403 | 335, 402,
336 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
404 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
405 | | nnre 10904 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
406 | 405 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 ∈ ℝ) |
407 | 392 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ∈ ℝ) |
408 | 394 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℝ) |
409 | 405 | ltp1d 10833 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1)) |
410 | 409 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < (𝑛 + 1)) |
411 | 398 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 + 1) ≤ 𝑚) |
412 | 406, 407,
408, 410, 411 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑛 < 𝑚) |
413 | 412 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → 𝑛 < 𝑚) |
414 | 406, 408 | ltnled 10063 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (𝑛 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑛)) |
415 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑚 = 𝑧 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛)) |
416 | 415 | equcoms 1934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑧 = 𝑚 → (𝑚 ≤ 𝑛 ↔ 𝑧 ≤ 𝑛)) |
417 | 416 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑧 = 𝑚 → (¬ 𝑚 ≤ 𝑛 ↔ ¬ 𝑧 ≤ 𝑛)) |
418 | 414, 417 | sylan9bb 732 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → (𝑛 < 𝑚 ↔ ¬ 𝑧 ≤ 𝑛)) |
419 | 413, 418 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ≤ 𝑛) |
420 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ≤ 𝑛) |
421 | 419, 420 | nsyl 134 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → ¬ 𝑧 ∈ (1...𝑛)) |
422 | 421 | iffalsed 4047 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = 〈0,
0〉) |
423 | 389 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℤ) |
424 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 ∈ ℝ) |
425 | 396 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < (𝑛 + 1)) |
426 | 424, 407,
408, 425, 411 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 0 < 𝑚) |
427 | 423, 426,
401 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 𝑚 ∈ ℕ) |
428 | 243 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → 〈0, 0〉 ∈
V) |
429 | 404, 422,
427, 428 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑛 ∈ ℕ ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = 〈0,
0〉) |
430 | 429 | ad4ant14 1285 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = 〈0,
0〉) |
431 | 430 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘〈0, 0〉)) |
432 | 403, 431 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘〈0, 0〉)) |
433 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ 〈0, 0〉 ∈
(ℂ × ℂ)) → ((abs ∘ − )‘〈0,
0〉) = (abs‘( − ‘〈0, 0〉))) |
434 | 139, 349,
433 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((abs
∘ − )‘〈0, 0〉) = (abs‘( −
‘〈0, 0〉)) |
435 | | df-ov 6552 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (0
− 0) = ( − ‘〈0, 0〉) |
436 | | 0m0e0 11007 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (0
− 0) = 0 |
437 | 435, 436 | eqtr3i 2634 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ( −
‘〈0, 0〉) = 0 |
438 | 437 | fveq2i 6106 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(abs‘( − ‘〈0, 0〉)) =
(abs‘0) |
439 | | abs0 13873 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(abs‘0) = 0 |
440 | 438, 439 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(abs‘( − ‘〈0, 0〉)) = 0 |
441 | 434, 440 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((abs
∘ − )‘〈0, 0〉) = 0 |
442 | 432, 441 | syl6eq 2660 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = 0) |
443 | | elfzuz 12209 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) → 𝑚 ∈ (ℤ≥‘(𝑛 + 1))) |
444 | | c0ex 9913 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ 0 ∈
V |
445 | 444 | fvconst2 6374 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑚 ∈
(ℤ≥‘(𝑛 + 1)) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
446 | 443, 445 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ((𝑛 + 1)...𝑡) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
447 | 446 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) →
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚) = 0) |
448 | 442, 447 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ((𝑛 + 1)...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) =
(((ℤ≥‘(𝑛 + 1)) × {0})‘𝑚)) |
449 | 381, 448 | seqfveq 12687 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡)) |
450 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(ℤ≥‘(𝑛 + 1)) = (ℤ≥‘(𝑛 + 1)) |
451 | 450 | ser0 12715 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈
(ℤ≥‘(𝑛 + 1)) → (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0) |
452 | 381, 451 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + ,
((ℤ≥‘(𝑛 + 1)) × {0}))‘𝑡) = 0) |
453 | 449, 452 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = 0) |
454 | 453 | adantlll 750 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = 0) |
455 | 388, 454 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡)) = ((seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛) + 0)) |
456 | 174 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) ∈ ℝ) |
457 | 328, 456 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
458 | 457 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
459 | | readdcl 9898 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑚 + 𝑣) ∈ ℝ) |
460 | 459 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ (𝑚 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (𝑚 + 𝑣) ∈ ℝ) |
461 | 316, 458,
460 | seqcl 12683 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) ∈ ℝ) |
462 | 461 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℝ) |
463 | 462 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℂ) |
464 | 463 | addid1d 10115 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) + 0) = (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛)) |
465 | 455, 464 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → ((seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑛) + (seq(𝑛 + 1)( + , ((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡)) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
466 | 387, 465 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
467 | 456 | ad5ant15 1295 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) ∈ ℝ) |
468 | 328, 467 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
469 | 383, 468,
367 | seqcl 12683 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ∈
ℝ) |
470 | 469 | leidd 10473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
471 | 466, 470 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑛 < 𝑡) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
472 | | elnnuz 11600 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑡 ∈ ℕ ↔ 𝑡 ∈
(ℤ≥‘1)) |
473 | 472 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
(ℤ≥‘1)) |
474 | 473 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑡 ∈
(ℤ≥‘1)) |
475 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)) = (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))) |
476 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 = 𝑚) |
477 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (1...𝑡) → 1 ≤ 𝑚) |
478 | 477 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 1 ≤ 𝑚) |
479 | 385 | nnred 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℝ) |
480 | 479 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℝ) |
481 | | nnre 10904 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑡 ∈ ℕ → 𝑡 ∈
ℝ) |
482 | 481 | ad3antlr 763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ∈ ℝ) |
483 | 405 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑛 ∈ ℝ) |
484 | | elfzle2 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ≤ 𝑡) |
485 | 484 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑡) |
486 | | simplr 788 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑡 ≤ 𝑛) |
487 | 480, 482,
483, 485, 486 | letrd 10073 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ≤ 𝑛) |
488 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ (1...𝑡) → 𝑚 ∈ ℤ) |
489 | 280 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ ℤ) |
490 | | elfz 12203 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑚 ∈ ℤ ∧ 1 ∈
ℤ ∧ 𝑛 ∈
ℤ) → (𝑚 ∈
(1...𝑛) ↔ (1 ≤
𝑚 ∧ 𝑚 ≤ 𝑛))) |
491 | 176, 490 | mp3an2 1404 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛))) |
492 | 488, 489,
491 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑚 ∈ (1...𝑛) ↔ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑛))) |
493 | 478, 487,
492 | mpbir2and 959 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛)) |
494 | 493 | ad5ant2345 1309 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ (1...𝑛)) |
495 | 494 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑚 ∈ (1...𝑛)) |
496 | 476, 495 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → 𝑧 ∈ (1...𝑛)) |
497 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ (1...𝑛) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑧)) |
498 | 496, 497 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑧)) |
499 | 240 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → (𝑓‘𝑧) = (𝑓‘𝑚)) |
500 | 498, 499 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) ∧ 𝑧 = 𝑚) → if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉) = (𝑓‘𝑚)) |
501 | 385 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → 𝑚 ∈ ℕ) |
502 | 242 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (𝑓‘𝑚) ∈ V) |
503 | 475, 500,
501, 502 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚) = (𝑓‘𝑚)) |
504 | 503 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚)) = ((abs ∘ −
)‘(𝑓‘𝑚))) |
505 | 335, 385,
336 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑚 ∈ (1...𝑡) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
506 | 505 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = ((abs ∘ −
)‘((𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))‘𝑚))) |
507 | | simplll 794 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) |
508 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
509 | 507, 385,
508 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) = ((abs ∘ − )‘(𝑓‘𝑚))) |
510 | 504, 506,
509 | 3eqtr4d 2654 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑡)) → (((abs ∘ − ) ∘
(𝑧 ∈ ℕ ↦
if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))‘𝑚) = (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
511 | 474, 510 | seqfveq 12687 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) = (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑡)) |
512 | | eluz 11577 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑡 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑛 ∈
(ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛)) |
513 | 377, 280,
512 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) → (𝑛 ∈
(ℤ≥‘𝑡) ↔ 𝑡 ≤ 𝑛)) |
514 | 513 | biimpar 501 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑛 ∈ ℕ ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡)) |
515 | 514 | adantlll 750 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → 𝑛 ∈ (ℤ≥‘𝑡)) |
516 | 507, 328,
456 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ (1...𝑛)) → (((abs ∘ − ) ∘
𝑓)‘𝑚) ∈ ℝ) |
517 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℤ) |
518 | 517 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℤ) |
519 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ∈ ℝ) |
520 | | peano2nn 10909 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈
ℕ) |
521 | 520 | nnred 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 ∈ ℕ → (𝑡 + 1) ∈
ℝ) |
522 | 521 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ∈ ℝ) |
523 | 517 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → 𝑚 ∈ ℝ) |
524 | 523 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℝ) |
525 | 520 | nngt0d 10941 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑡 ∈ ℕ → 0 <
(𝑡 + 1)) |
526 | 525 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < (𝑡 + 1)) |
527 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑚 ∈ ((𝑡 + 1)...𝑛) → (𝑡 + 1) ≤ 𝑚) |
528 | 527 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → (𝑡 + 1) ≤ 𝑚) |
529 | 519, 522,
524, 526, 528 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 < 𝑚) |
530 | 518, 529,
401 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑡 ∈ ℕ ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
531 | 530 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑡 ∈ ℕ ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
532 | 531 | adantlll 750 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 𝑚 ∈ ℕ) |
533 | 172 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (𝑓‘𝑚) ∈ (ℂ ×
ℂ)) |
534 | | ffvelrn 6265 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) → (
− ‘(𝑓‘𝑚)) ∈ ℂ) |
535 | 139, 533,
534 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ( −
‘(𝑓‘𝑚)) ∈
ℂ) |
536 | 535 | absge0d 14031 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (abs‘(
− ‘(𝑓‘𝑚)))) |
537 | | fvco3 6185 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((
− :(ℂ × ℂ)⟶ℂ ∧ (𝑓‘𝑚) ∈ (ℂ × ℂ)) →
((abs ∘ − )‘(𝑓‘𝑚)) = (abs‘( − ‘(𝑓‘𝑚)))) |
538 | 139, 533,
537 | sylancr 694 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → ((abs ∘ −
)‘(𝑓‘𝑚)) = (abs‘( −
‘(𝑓‘𝑚)))) |
539 | 508, 538 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝑓)‘𝑚) = (abs‘( − ‘(𝑓‘𝑚)))) |
540 | 536, 539 | breqtrrd 4611 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝑓)‘𝑚)) |
541 | 540 | ad5ant15 1295 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝑓)‘𝑚)) |
542 | 532, 541 | syldan 486 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) ∧ 𝑚 ∈ ((𝑡 + 1)...𝑛)) → 0 ≤ (((abs ∘ − )
∘ 𝑓)‘𝑚)) |
543 | 474, 515,
516, 542 | sermono 12695 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
544 | 511, 543 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ≤ 𝑛) → (seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
545 | 405 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑛 ∈ ℝ) |
546 | 481 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → 𝑡 ∈ ℝ) |
547 | 471, 544,
545, 546 | ltlecasei 10024 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑡 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
548 | 547 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
549 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 = (seq1( + , ((abs ∘
− ) ∘ (𝑧 ∈
ℕ ↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) → (𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
550 | 549 | ralrn 6270 |
. . . . . . . . . . . . . . . . . . 19
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))) Fn ℕ →
(∀𝑚 ∈ ran seq1(
+ , ((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
551 | 357, 550 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∀𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
552 | 551 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (∀𝑚 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ ∀𝑡 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))‘𝑡) ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛))) |
553 | 548, 552 | mpbird 246 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∀𝑚 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))𝑚 ≤ (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛)) |
554 | 553 | r19.21bi 2916 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → 𝑚 ≤ (seq1( + , ((abs ∘
− ) ∘ 𝑓))‘𝑛)) |
555 | 364, 365,
554 | lensymd 10067 |
. . . . . . . . . . . . . 14
⊢ (((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ ran seq1( + , ((abs ∘ − )
∘ (𝑧 ∈ ℕ
↦ if(𝑧 ∈
(1...𝑛), (𝑓‘𝑧), 〈0, 0〉))))) → ¬ (seq1(
+ , ((abs ∘ − ) ∘ 𝑓))‘𝑛) < 𝑚) |
556 | 313, 324,
360, 555 | supmax 8256 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)) |
557 | 52, 556 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝑧 ∈ ℕ ↦ if(𝑧 ∈ (1...𝑛), (𝑓‘𝑧), 〈0, 0〉)))), ℝ*,
< ) = (seq1( + , ((abs ∘ − ) ∘ 𝑓))‘𝑛)) |
558 | 257, 311,
557 | 3eqtr3rd 2653 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)))) |
559 | | elfznn 12241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 ∈ (1...𝑛) → 𝑧 ∈ ℕ) |
560 | 166, 65 | sseldi 3566 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → (𝑓‘𝑧) ∈ (ℝ ×
ℝ)) |
561 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(𝑓‘𝑧) = 〈(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))〉) |
562 | 561 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) =
([,]‘〈(1st ‘(𝑓‘𝑧)), (2nd ‘(𝑓‘𝑧))〉)) |
563 | | df-ov 6552 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) = ([,]‘〈(1st
‘(𝑓‘𝑧)), (2nd
‘(𝑓‘𝑧))〉) |
564 | 562, 563 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) = ((1st
‘(𝑓‘𝑧))[,](2nd
‘(𝑓‘𝑧)))) |
565 | | xp1st 7089 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(1st ‘(𝑓‘𝑧)) ∈ ℝ) |
566 | | xp2nd 7090 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
(2nd ‘(𝑓‘𝑧)) ∈ ℝ) |
567 | | iccssre 12126 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd
‘(𝑓‘𝑧)) ∈ ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ) |
568 | 565, 566,
567 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ⊆ ℝ) |
569 | 564, 568 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) ⊆
ℝ) |
570 | 560, 569 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
571 | 52, 559, 570 | syl2an 493 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
572 | 571 | ralrimiva 2949 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
573 | | iunss 4497 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
574 | 572, 573 | sylibr 223 |
. . . . . . . . . . . . 13
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
575 | 574 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ) |
576 | | uzid 11578 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 + 1) ∈ ℤ →
(𝑛 + 1) ∈
(ℤ≥‘(𝑛 + 1))) |
577 | | ne0i 3880 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 + 1) ∈
(ℤ≥‘(𝑛 + 1)) →
(ℤ≥‘(𝑛 + 1)) ≠ ∅) |
578 | | iunconst 4465 |
. . . . . . . . . . . . . . . 16
⊢
((ℤ≥‘(𝑛 + 1)) ≠ ∅ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = ([,]‘〈0, 0〉)) |
579 | 376, 576,
577, 578 | 4syl 19 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = ([,]‘〈0, 0〉)) |
580 | | iccid 12091 |
. . . . . . . . . . . . . . . . 17
⊢ (0 ∈
ℝ* → (0[,]0) = {0}) |
581 | 261, 580 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ (0[,]0) =
{0} |
582 | | df-ov 6552 |
. . . . . . . . . . . . . . . 16
⊢ (0[,]0) =
([,]‘〈0, 0〉) |
583 | 581, 582 | eqtr3i 2634 |
. . . . . . . . . . . . . . 15
⊢ {0} =
([,]‘〈0, 0〉) |
584 | 579, 583 | syl6eqr 2662 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) = {0}) |
585 | | snssi 4280 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
ℝ → {0} ⊆ ℝ) |
586 | 201, 585 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ {0}
⊆ ℝ |
587 | 584, 586 | syl6eqss 3618 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ) |
588 | 587 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ) |
589 | 584 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ →
(vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = (vol*‘{0})) |
590 | 589 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = (vol*‘{0})) |
591 | | ovolsn 23070 |
. . . . . . . . . . . . . 14
⊢ (0 ∈
ℝ → (vol*‘{0}) = 0) |
592 | 201, 591 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(vol*‘{0}) = 0 |
593 | 590, 592 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉)) = 0) |
594 | | ovolunnul 23075 |
. . . . . . . . . . . 12
⊢
((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ ℝ ∧ ∪ 𝑧 ∈ (ℤ≥‘(𝑛 + 1))([,]‘〈0,
0〉) ⊆ ℝ ∧ (vol*‘∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉)) = 0)
→ (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉))) =
(vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
595 | 575, 588,
593, 594 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (vol*‘(∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∪ ∪
𝑧 ∈
(ℤ≥‘(𝑛 + 1))([,]‘〈0, 0〉))) =
(vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
596 | 558, 595 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝑓))‘𝑛) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
597 | 596 | breq2d 4595 |
. . . . . . . . 9
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
598 | 597 | biimpd 218 |
. . . . . . . 8
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑛 ∈ ℕ) → (𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
599 | 598 | reximdva 3000 |
. . . . . . 7
⊢ (𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
600 | 599 | adantl 481 |
. . . . . 6
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → (∃𝑛 ∈ ℕ 𝑀 < (seq1( + , ((abs ∘ − )
∘ 𝑓))‘𝑛) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
601 | 196, 600 | mpd 15 |
. . . . 5
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑛 ∈ ℕ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
602 | | fzfi 12633 |
. . . . . . . . . 10
⊢
(1...𝑛) ∈
Fin |
603 | | icccld 22380 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘(𝑓‘𝑧)) ∈ ℝ ∧ (2nd
‘(𝑓‘𝑧)) ∈ ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran
(,)))) |
604 | 565, 566,
603 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
((1st ‘(𝑓‘𝑧))[,](2nd ‘(𝑓‘𝑧))) ∈ (Clsd‘(topGen‘ran
(,)))) |
605 | 564, 604 | eqeltrd 2688 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ∈ (ℝ × ℝ) →
([,]‘(𝑓‘𝑧)) ∈
(Clsd‘(topGen‘ran (,)))) |
606 | 560, 605 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
607 | 559, 606 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
608 | 607 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
609 | | uniretop 22376 |
. . . . . . . . . . 11
⊢ ℝ =
∪ (topGen‘ran (,)) |
610 | 609 | iuncld 20659 |
. . . . . . . . . 10
⊢
(((topGen‘ran (,)) ∈ Top ∧ (1...𝑛) ∈ Fin ∧ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
611 | 1, 602, 608, 610 | mp3an12i 1420 |
. . . . . . . . 9
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
612 | 611 | adantr 480 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran
(,)))) |
613 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = (𝑓‘𝑧) → ([,]‘𝑏) = ([,]‘(𝑓‘𝑧))) |
614 | 613 | sseq1d 3595 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑓‘𝑧) → (([,]‘𝑏) ⊆ 𝐴 ↔ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
615 | 614 | elrab 3331 |
. . . . . . . . . . . . . 14
⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ↔ ((𝑓‘𝑧) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∧ ([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
616 | 615 | simprbi 479 |
. . . . . . . . . . . . 13
⊢ ((𝑓‘𝑧) ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
617 | 65, 73, 616 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ ℕ) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
618 | 559, 617 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ 𝑧 ∈ (1...𝑛)) → ([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
619 | 618 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
620 | | iunss 4497 |
. . . . . . . . . 10
⊢ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ↔ ∀𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
621 | 619, 620 | sylibr 223 |
. . . . . . . . 9
⊢ (𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} → ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
622 | 621 | adantr 480 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴) |
623 | | simprr 792 |
. . . . . . . 8
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
624 | | sseq1 3589 |
. . . . . . . . . 10
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑠 ⊆ 𝐴 ↔ ∪
𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴)) |
625 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (vol*‘𝑠) = (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))) |
626 | 625 | breq2d 4595 |
. . . . . . . . . 10
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → (𝑀 < (vol*‘𝑠) ↔ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) |
627 | 624, 626 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑠 = ∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) → ((𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)) ↔ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)))))) |
628 | 627 | rspcev 3282 |
. . . . . . . 8
⊢
((∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ∈ (Clsd‘(topGen‘ran (,)))
∧ (∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧)) ⊆ 𝐴 ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
629 | 612, 622,
623, 628 | syl12anc 1316 |
. . . . . . 7
⊢ ((𝑓:ℕ⟶{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
630 | 52, 629 | sylan 487 |
. . . . . 6
⊢ ((𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)} ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
631 | 630 | adantll 746 |
. . . . 5
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) ∧ (𝑛 ∈ ℕ ∧ 𝑀 < (vol*‘∪ 𝑧 ∈ (1...𝑛)([,]‘(𝑓‘𝑧))))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
632 | 601, 631 | rexlimddv 3017 |
. . . 4
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
633 | 632 | adantlr 747 |
. . 3
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) ∧ 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦
〈(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))〉) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)}) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
634 | 17, 633 | exlimddv 1850 |
. 2
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) ∧ 𝐴 ≠ ∅) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |
635 | 15, 634 | pm2.61dane 2869 |
1
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑀 ∈ ℝ
∧ 𝑀 <
(vol*‘𝐴)) →
∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠))) |