Step | Hyp | Ref
| Expression |
1 | | caratheodorylem1.n |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | eluzfz2 12220 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
4 | | id 22 |
. 2
⊢ (𝜑 → 𝜑) |
5 | | fveq2 6103 |
. . . . . 6
⊢ (𝑗 = 𝑀 → (𝐺‘𝑗) = (𝐺‘𝑀)) |
6 | 5 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑀 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑀))) |
7 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀)) |
8 | 7 | mpteq1d 4666 |
. . . . . 6
⊢ (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))) |
9 | 8 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑀 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))) |
10 | 6, 9 | eqeq12d 2625 |
. . . 4
⊢ (𝑗 = 𝑀 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))) |
11 | 10 | imbi2d 329 |
. . 3
⊢ (𝑗 = 𝑀 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))))) |
12 | | fveq2 6103 |
. . . . . 6
⊢ (𝑗 = 𝑖 → (𝐺‘𝑗) = (𝐺‘𝑖)) |
13 | 12 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑖 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑖))) |
14 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖)) |
15 | 14 | mpteq1d 4666 |
. . . . . 6
⊢ (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) |
16 | 15 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑖 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
17 | 13, 16 | eqeq12d 2625 |
. . . 4
⊢ (𝑗 = 𝑖 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))) |
18 | 17 | imbi2d 329 |
. . 3
⊢ (𝑗 = 𝑖 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))))) |
19 | | fveq2 6103 |
. . . . . 6
⊢ (𝑗 = (𝑖 + 1) → (𝐺‘𝑗) = (𝐺‘(𝑖 + 1))) |
20 | 19 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1)))) |
21 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1))) |
22 | 21 | mpteq1d 4666 |
. . . . . 6
⊢ (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) |
23 | 22 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = (𝑖 + 1) →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))) |
24 | 20, 23 | eqeq12d 2625 |
. . . 4
⊢ (𝑗 = (𝑖 + 1) → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘(𝑖 + 1))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))))) |
25 | 24 | imbi2d 329 |
. . 3
⊢ (𝑗 = (𝑖 + 1) → ((𝜑 → (𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))))) |
26 | | fveq2 6103 |
. . . . . 6
⊢ (𝑗 = 𝑁 → (𝐺‘𝑗) = (𝐺‘𝑁)) |
27 | 26 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑁 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑁))) |
28 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁)) |
29 | 28 | mpteq1d 4666 |
. . . . . 6
⊢ (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))) |
30 | 29 | fveq2d 6107 |
. . . . 5
⊢ (𝑗 = 𝑁 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) |
31 | 27, 30 | eqeq12d 2625 |
. . . 4
⊢ (𝑗 = 𝑁 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))) |
32 | 31 | imbi2d 329 |
. . 3
⊢ (𝑗 = 𝑁 → ((𝜑 → (𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))))) ↔ (𝜑 → (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))))) |
33 | | eluzel2 11568 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
34 | 1, 33 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
35 | | fzsn 12254 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
36 | 34, 35 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
37 | 36 | mpteq1d 4666 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) |
38 | 37 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))))) |
39 | | caratheodorylem1.o |
. . . . . . . . 9
⊢ (𝜑 → 𝑂 ∈ OutMeas) |
40 | 39 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas) |
41 | | eqid 2610 |
. . . . . . . 8
⊢ ∪ dom 𝑂 = ∪ dom 𝑂 |
42 | | caratheodorylem1.s |
. . . . . . . . . . . 12
⊢ 𝑆 = (CaraGen‘𝑂) |
43 | 42 | caragenss 39394 |
. . . . . . . . . . 11
⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
44 | 40, 43 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂) |
45 | | caratheodorylem1.e |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸:𝑍⟶𝑆) |
46 | 45 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝐸:𝑍⟶𝑆) |
47 | | elsni 4142 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ {𝑀} → 𝑛 = 𝑀) |
48 | 47 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 = 𝑀) |
49 | | uzid 11578 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
50 | 34, 49 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
51 | | caratheodorylem1.z |
. . . . . . . . . . . . . 14
⊢ 𝑍 =
(ℤ≥‘𝑀) |
52 | 50, 51 | syl6eleqr 2699 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ 𝑍) |
53 | 52 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑀 ∈ 𝑍) |
54 | 48, 53 | eqeltrd 2688 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 ∈ 𝑍) |
55 | 46, 54 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ 𝑆) |
56 | 44, 55 | sseldd 3569 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ dom 𝑂) |
57 | | elssuni 4403 |
. . . . . . . . 9
⊢ ((𝐸‘𝑛) ∈ dom 𝑂 → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
58 | 56, 57 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
59 | 40, 41, 58 | omecl 39393 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) |
60 | | eqid 2610 |
. . . . . . 7
⊢ (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) |
61 | 59, 60 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))):{𝑀}⟶(0[,]+∞)) |
62 | 34, 61 | sge0sn 39272 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀)) |
63 | | eqidd 2611 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) |
64 | 36 | iuneq1d 4481 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖)) |
65 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑀 → (𝐸‘𝑖) = (𝐸‘𝑀)) |
66 | 65 | iunxsng 4538 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ 𝑍 → ∪
𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀)) |
67 | 52, 66 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀)) |
68 | | eqidd 2611 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐸‘𝑀) = (𝐸‘𝑀)) |
69 | 64, 67, 68 | 3eqtrrd 2649 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
70 | 69 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
71 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑛 = 𝑀 → (𝐸‘𝑛) = (𝐸‘𝑀)) |
72 | 71 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐸‘𝑀)) |
73 | | caratheodorylem1.g |
. . . . . . . . . . 11
⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) |
74 | 73 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖))) |
75 | | oveq2 6557 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀)) |
76 | 75 | iuneq1d 4481 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑀 → ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
77 | 76 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
78 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ (𝑀...𝑀) ∈ V |
79 | | fvex 6113 |
. . . . . . . . . . . 12
⊢ (𝐸‘𝑖) ∈ V |
80 | 78, 79 | iunex 7039 |
. . . . . . . . . . 11
⊢ ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V |
81 | 80 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V) |
82 | 74, 77, 52, 81 | fvmptd 6197 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
83 | 82 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐺‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
84 | 70, 72, 83 | 3eqtr4d 2654 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐺‘𝑀)) |
85 | 84 | fveq2d 6107 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐺‘𝑀))) |
86 | | snidg 4153 |
. . . . . . 7
⊢ (𝑀 ∈ 𝑍 → 𝑀 ∈ {𝑀}) |
87 | 52, 86 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ {𝑀}) |
88 | | fvex 6113 |
. . . . . . 7
⊢ (𝑂‘(𝐺‘𝑀)) ∈ V |
89 | 88 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) ∈ V) |
90 | 63, 85, 87, 89 | fvmptd 6197 |
. . . . 5
⊢ (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀) = (𝑂‘(𝐺‘𝑀))) |
91 | 38, 62, 90 | 3eqtrrd 2649 |
. . . 4
⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))) |
92 | 91 | a1i 11 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))) |
93 | | simp3 1056 |
. . . . 5
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝜑) |
94 | | simp1 1054 |
. . . . 5
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁)) |
95 | | id 22 |
. . . . . . 7
⊢ ((𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))) |
96 | 95 | imp 444 |
. . . . . 6
⊢ (((𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
97 | 96 | 3adant1 1072 |
. . . . 5
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
98 | | elfzoel1 12337 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) |
99 | | elfzoelz 12339 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ) |
100 | 99 | peano2zd 11361 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ) |
101 | 98, 100, 100 | 3jca 1235 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈
ℤ)) |
102 | 98 | zred 11358 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ) |
103 | 100 | zred 11358 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ) |
104 | 99 | zred 11358 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ) |
105 | | elfzole1 12347 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑖) |
106 | 104 | ltp1d 10833 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1)) |
107 | 102, 104,
103, 105, 106 | lelttrd 10074 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1)) |
108 | 102, 103,
107 | ltled 10064 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1)) |
109 | | leid 10012 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 + 1) ∈ ℝ →
(𝑖 + 1) ≤ (𝑖 + 1)) |
110 | 103, 109 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1)) |
111 | 101, 108,
110 | jca32 556 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧
(𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1)))) |
112 | | elfz2 12204 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ) ∧
(𝑀 ≤ (𝑖 + 1) ∧ (𝑖 + 1) ≤ (𝑖 + 1)))) |
113 | 111, 112 | sylibr 223 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1))) |
114 | 113 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1))) |
115 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑖 + 1) → (𝐸‘𝑗) = (𝐸‘(𝑖 + 1))) |
116 | 115 | ssiun2s 4500 |
. . . . . . . . . . . 12
⊢ ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
117 | 114, 116 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
118 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗)) |
119 | 118 | cbviunv 4495 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) |
120 | 119 | mpteq2i 4669 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 ↦ ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)) |
121 | 73, 120 | eqtri 2632 |
. . . . . . . . . . . . . 14
⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)) |
122 | 121 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗))) |
123 | | oveq2 6557 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1))) |
124 | 123 | iuneq1d 4481 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
125 | 124 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 = (𝑖 + 1)) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
126 | 34 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ) |
127 | 99 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ) |
128 | 127 | peano2zd 11361 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ) |
129 | 126 | zred 11358 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ) |
130 | 128 | zred 11358 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ) |
131 | 127 | zred 11358 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ) |
132 | 105 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑖) |
133 | 131 | ltp1d 10833 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1)) |
134 | 129, 131,
130, 132, 133 | lelttrd 10074 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1)) |
135 | 129, 130,
134 | ltled 10064 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1)) |
136 | 126, 128,
135 | 3jca 1235 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1))) |
137 | | eluz2 11569 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 + 1) ∈
(ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1))) |
138 | 136, 137 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈
(ℤ≥‘𝑀)) |
139 | 51 | eqcomi 2619 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑀) = 𝑍 |
140 | 138, 139 | syl6eleq 2698 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍) |
141 | | ovex 6577 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...(𝑖 + 1)) ∈ V |
142 | | fvex 6113 |
. . . . . . . . . . . . . . 15
⊢ (𝐸‘𝑗) ∈ V |
143 | 141, 142 | iunex 7039 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V |
144 | 143 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V) |
145 | 122, 125,
140, 144 | fvmptd 6197 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ∪
𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
146 | 145 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (𝐺‘(𝑖 + 1))) |
147 | 117, 146 | sseqtrd 3604 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1))) |
148 | | sseqin2 3779 |
. . . . . . . . . . 11
⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
149 | 148 | biimpi 205 |
. . . . . . . . . 10
⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
150 | 147, 149 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
151 | 150 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1)))) |
152 | | nfcv 2751 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗(𝐸‘(𝑖 + 1)) |
153 | | elfzouz 12343 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ≥‘𝑀)) |
154 | 153 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ≥‘𝑀)) |
155 | 152, 154,
115 | iunp1 38260 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1)))) |
156 | 145, 155 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = (∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1)))) |
157 | 156 | difeq1d 3689 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1)))) |
158 | | caratheodorylem1.dj |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
159 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗)) |
160 | 159 | cbvdisjv 4564 |
. . . . . . . . . . . . . . 15
⊢
(Disj 𝑛
∈ 𝑍 (𝐸‘𝑛) ↔ Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
161 | 158, 160 | sylib 207 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
162 | 161 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
163 | | fzssuz 12253 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...𝑖) ⊆ (ℤ≥‘𝑀) |
164 | 163, 139 | sseqtri 3600 |
. . . . . . . . . . . . . 14
⊢ (𝑀...𝑖) ⊆ 𝑍 |
165 | 164 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍) |
166 | | fzp1nel 12293 |
. . . . . . . . . . . . . . . 16
⊢ ¬
(𝑖 + 1) ∈ (𝑀...𝑖) |
167 | 166 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖)) |
168 | 167 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖)) |
169 | 140, 168 | eldifd 3551 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖))) |
170 | 162, 165,
169, 115 | disjiun2 38251 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅) |
171 | | undif4 3987 |
. . . . . . . . . . . 12
⊢
((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅ → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1)))) |
172 | 170, 171 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1)))) |
173 | 172 | eqcomd 2616 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1))) = (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) |
174 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝜑) |
175 | 154, 139 | syl6eleq 2698 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ 𝑍) |
176 | 121 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗))) |
177 | | simpr 476 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖) |
178 | 177 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖)) |
179 | 178 | iuneq1d 4481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
180 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍) |
181 | | ovex 6577 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀...𝑖) ∈ V |
182 | 181, 142 | iunex 7039 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V |
183 | 182 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V) |
184 | 176, 179,
180, 183 | fvmptd 6197 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
185 | 174, 175,
184 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
186 | 185 | eqcomd 2616 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) = (𝐺‘𝑖)) |
187 | | difid 3902 |
. . . . . . . . . . . . 13
⊢ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅ |
188 | 187 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅) |
189 | 186, 188 | uneq12d 3730 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺‘𝑖) ∪ ∅)) |
190 | | un0 3919 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖) |
191 | 190 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖)) |
192 | 189, 191 | eqtrd 2644 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺‘𝑖)) |
193 | 157, 173,
192 | 3eqtrd 2648 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺‘𝑖)) |
194 | 193 | fveq2d 6107 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺‘𝑖))) |
195 | 151, 194 | oveq12d 6567 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
196 | 195 | 3adant3 1074 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
197 | 39 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas) |
198 | 45 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍⟶𝑆) |
199 | 198, 140 | ffvelrnd 6268 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆) |
200 | | simpll 786 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑) |
201 | 98 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ) |
202 | | elfzelz 12213 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ) |
203 | 202 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ) |
204 | | elfzle1 12215 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀 ≤ 𝑗) |
205 | 204 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ≤ 𝑗) |
206 | 201, 203,
205 | 3jca 1235 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) |
207 | | eluz2 11569 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) |
208 | 206, 207 | sylibr 223 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ≥‘𝑀)) |
209 | 208, 139 | syl6eleq 2698 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍) |
210 | 209 | adantll 746 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍) |
211 | 39, 43 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑆 ⊆ dom 𝑂) |
212 | 211 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑆 ⊆ dom 𝑂) |
213 | 45 | ffvelrnda 6267 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ 𝑆) |
214 | 212, 213 | sseldd 3569 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑂) |
215 | | elssuni 4403 |
. . . . . . . . . . . . . 14
⊢ ((𝐸‘𝑗) ∈ dom 𝑂 → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
216 | 214, 215 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
217 | 200, 210,
216 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
218 | 217 | ralrimiva 2949 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
219 | | iunss 4497 |
. . . . . . . . . . 11
⊢ (∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
220 | 218, 219 | sylibr 223 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
221 | 145, 220 | eqsstrd 3602 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ ∪
dom 𝑂) |
222 | 197, 42, 41, 199, 221 | caragensplit 39390 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1)))) |
223 | 222 | eqcomd 2616 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))) |
224 | 223 | 3adant3 1074 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))) |
225 | 197 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas) |
226 | 174 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑) |
227 | | elfzuz 12209 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
228 | 227, 139 | syl6eleq 2698 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ 𝑍) |
229 | 228 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛 ∈ 𝑍) |
230 | 45, 211 | fssd 5970 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑂) |
231 | 230 | ffvelrnda 6267 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑂) |
232 | 231, 57 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
233 | 226, 229,
232 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
234 | 225, 41, 233 | omecl 39393 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) |
235 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑖 + 1) → (𝐸‘𝑛) = (𝐸‘(𝑖 + 1))) |
236 | 235 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1)))) |
237 | 154, 234,
236 | sge0p1 39307 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) →
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) =
((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1))))) |
238 | 237 | 3adant3 1074 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) →
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) =
((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1))))) |
239 | | id 22 |
. . . . . . . . . 10
⊢ ((𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
240 | 239 | eqcomd 2616 |
. . . . . . . . 9
⊢ ((𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) →
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑖))) |
241 | 240 | oveq1d 6564 |
. . . . . . . 8
⊢ ((𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) →
((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺‘𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1))))) |
242 | 241 | 3ad2ant3 1077 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) →
((Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐺‘𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1))))) |
243 | | simpl 472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝜑) |
244 | 164 | sseli 3564 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (𝑀...𝑖) → 𝑗 ∈ 𝑍) |
245 | 244 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝑗 ∈ 𝑍) |
246 | 243, 245,
216 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
247 | 246 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
248 | 247 | ralrimiva 2949 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
249 | | iunss 4497 |
. . . . . . . . . . . . 13
⊢ (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
250 | 248, 249 | sylibr 223 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
251 | 184, 250 | eqsstrd 3602 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ⊆ ∪ dom
𝑂) |
252 | 174, 175,
251 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) ⊆ ∪ dom
𝑂) |
253 | 197, 41, 252 | omexrcl 39397 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘𝑖)) ∈
ℝ*) |
254 | 117, 220 | sstrd 3578 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪
dom 𝑂) |
255 | 197, 41, 254 | omexrcl 39397 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈
ℝ*) |
256 | 253, 255 | xaddcomd 38481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺‘𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
257 | 256 | 3adant3 1074 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → ((𝑂‘(𝐺‘𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
258 | 238, 242,
257 | 3eqtrd 2648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) →
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
259 | 196, 224,
258 | 3eqtr4d 2654 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁) ∧ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝑂‘(𝐺‘(𝑖 + 1))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))) |
260 | 93, 94, 97, 259 | syl3anc 1318 |
. . . 4
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘(𝑖 + 1))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))) |
261 | 260 | 3exp 1256 |
. . 3
⊢ (𝑖 ∈ (𝑀..^𝑁) → ((𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) → (𝜑 → (𝑂‘(𝐺‘(𝑖 + 1))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))))) |
262 | 11, 18, 25, 32, 92, 261 | fzind2 12448 |
. 2
⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))) |
263 | 3, 4, 262 | sylc 63 |
1
⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) |