Step | Hyp | Ref
| Expression |
1 | | 3simpb 1052 |
. . 3
⊢ ((𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) |
2 | 1 | reximi 2994 |
. 2
⊢
(∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) → ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) |
3 | | fveq2 6103 |
. . . . . 6
⊢ (𝑚 = 𝑤 → (ℤ≥‘𝑚) =
(ℤ≥‘𝑤)) |
4 | 3 | sseq2d 3596 |
. . . . 5
⊢ (𝑚 = 𝑤 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑤))) |
5 | | seqeq1 12666 |
. . . . . 6
⊢ (𝑚 = 𝑤 → seq𝑚( · , 𝐹) = seq𝑤( · , 𝐹)) |
6 | 5 | breq1d 4593 |
. . . . 5
⊢ (𝑚 = 𝑤 → (seq𝑚( · , 𝐹) ⇝ 𝑥 ↔ seq𝑤( · , 𝐹) ⇝ 𝑥)) |
7 | 4, 6 | anbi12d 743 |
. . . 4
⊢ (𝑚 = 𝑤 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥))) |
8 | 7 | cbvrexv 3148 |
. . 3
⊢
(∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥) ↔ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) |
9 | | reeanv 3086 |
. . . . 5
⊢
(∃𝑤 ∈
ℤ ∃𝑚 ∈
ℕ ((𝐴 ⊆
(ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) ↔ (∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)))) |
10 | | simprlr 799 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑤( · , 𝐹) ⇝ 𝑥) |
11 | | simprll 798 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑤)) |
12 | | uzssz 11583 |
. . . . . . . . . . . . . . . . 17
⊢
(ℤ≥‘𝑤) ⊆ ℤ |
13 | | zssre 11261 |
. . . . . . . . . . . . . . . . 17
⊢ ℤ
⊆ ℝ |
14 | 12, 13 | sstri 3577 |
. . . . . . . . . . . . . . . 16
⊢
(ℤ≥‘𝑤) ⊆ ℝ |
15 | 11, 14 | syl6ss 3580 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℝ) |
16 | | ltso 9997 |
. . . . . . . . . . . . . . 15
⊢ < Or
ℝ |
17 | | soss 4977 |
. . . . . . . . . . . . . . 15
⊢ (𝐴 ⊆ ℝ → ( <
Or ℝ → < Or 𝐴)) |
18 | 15, 16, 17 | mpisyl 21 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → < Or 𝐴) |
19 | | fzfi 12633 |
. . . . . . . . . . . . . . 15
⊢
(1...𝑚) ∈
Fin |
20 | | ovex 6577 |
. . . . . . . . . . . . . . . . . 18
⊢
(1...𝑚) ∈
V |
21 | 20 | f1oen 7862 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → (1...𝑚) ≈ 𝐴) |
22 | 21 | ad2antll 761 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴) |
23 | 22 | ensymd 7893 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚)) |
24 | | enfii 8062 |
. . . . . . . . . . . . . . 15
⊢
(((1...𝑚) ∈ Fin
∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin) |
25 | 19, 23, 24 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin) |
26 | | fz1iso 13103 |
. . . . . . . . . . . . . 14
⊢ (( <
Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴)) |
27 | 18, 25, 26 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴)) |
28 | | prodmo.1 |
. . . . . . . . . . . . . . . 16
⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
29 | | simpll 786 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝜑) |
30 | | prodmo.2 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
31 | 29, 30 | sylan 487 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
32 | | prodmo.3 |
. . . . . . . . . . . . . . . 16
⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵) |
33 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ ↦
⦋(𝑔‘𝑗) / 𝑘⦌𝐵) = (𝑗 ∈ ℕ ↦ ⦋(𝑔‘𝑗) / 𝑘⦌𝐵) |
34 | | simplrr 797 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ) |
35 | | simplrl 796 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑤 ∈ ℤ) |
36 | | simplll 794 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆
(ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑤)) |
37 | 36 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑤)) |
38 | | simprlr 799 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴) |
39 | | simprr 792 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴)) |
40 | 28, 31, 32, 33, 34, 35, 37, 38, 39 | prodmolem2a 14503 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴))) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)) |
41 | 40 | expr 641 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))) |
42 | 41 | exlimdv 1848 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(#‘𝐴)), 𝐴) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚))) |
43 | 27, 42 | mpd 15 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)) |
44 | | climuni 14131 |
. . . . . . . . . . . 12
⊢
((seq𝑤( · ,
𝐹) ⇝ 𝑥 ∧ seq𝑤( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑚)) → 𝑥 = (seq1( · , 𝐺)‘𝑚)) |
45 | 10, 43, 44 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( · , 𝐺)‘𝑚)) |
46 | | eqeq2 2621 |
. . . . . . . . . . 11
⊢ (𝑧 = (seq1( · , 𝐺)‘𝑚) → (𝑥 = 𝑧 ↔ 𝑥 = (seq1( · , 𝐺)‘𝑚))) |
47 | 45, 46 | syl5ibrcom 236 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧)) |
48 | 47 | expr 641 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (𝑓:(1...𝑚)–1-1-onto→𝐴 → (𝑧 = (seq1( · , 𝐺)‘𝑚) → 𝑥 = 𝑧))) |
49 | 48 | impd 446 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧)) |
50 | 49 | exlimdv 1848 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) ∧ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧)) |
51 | 50 | expimpd 627 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑤 ∈ ℤ ∧ 𝑚 ∈ ℕ)) → (((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧)) |
52 | 51 | rexlimdvva 3020 |
. . . . 5
⊢ (𝜑 → (∃𝑤 ∈ ℤ ∃𝑚 ∈ ℕ ((𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧)) |
53 | 9, 52 | syl5bir 232 |
. . . 4
⊢ (𝜑 → ((∃𝑤 ∈ ℤ (𝐴 ⊆
(ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥) ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚))) → 𝑥 = 𝑧)) |
54 | 53 | expdimp 452 |
. . 3
⊢ ((𝜑 ∧ ∃𝑤 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑤) ∧ seq𝑤( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧)) |
55 | 8, 54 | sylan2b 491 |
. 2
⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧)) |
56 | 2, 55 | sylan2 490 |
1
⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈
(ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧)) |