| Step | Hyp | Ref
| Expression |
| 1 | | caratheodorylem1.n |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 2 | | eluzfz2 13572 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) |
| 3 | 1, 2 | syl 17 |
. 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) |
| 4 | | id 22 |
. 2
⊢ (𝜑 → 𝜑) |
| 5 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑗 = 𝑀 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑀))) |
| 6 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑗 = 𝑀 → (𝑀...𝑗) = (𝑀...𝑀)) |
| 7 | 6 | mpteq1d 5237 |
. . . . . 6
⊢ (𝑗 = 𝑀 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))) |
| 8 | 7 | fveq2d 6910 |
. . . . 5
⊢ (𝑗 = 𝑀 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 9 | 5, 8 | eqeq12d 2753 |
. . . 4
⊢ (𝑗 = 𝑀 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))) |
| 10 | 9 | imbi2d 340 |
. . 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))))))) |
| 11 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑗 = 𝑖 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑖))) |
| 12 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑗 = 𝑖 → (𝑀...𝑗) = (𝑀...𝑖)) |
| 13 | 12 | mpteq1d 5237 |
. . . . . 6
⊢ (𝑗 = 𝑖 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) |
| 14 | 13 | fveq2d 6910 |
. . . . 5
⊢ (𝑗 = 𝑖 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 15 | 11, 14 | eqeq12d 2753 |
. . . 4
⊢ (𝑗 = 𝑖 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))))) |
| 16 | 15 | imbi2d 340 |
. . 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))))))) |
| 17 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑗 = (𝑖 + 1) → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘(𝑖 + 1)))) |
| 18 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑗 = (𝑖 + 1) → (𝑀...𝑗) = (𝑀...(𝑖 + 1))) |
| 19 | 18 | mpteq1d 5237 |
. . . . . 6
⊢ (𝑗 = (𝑖 + 1) → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛)))) |
| 20 | 19 | fveq2d 6910 |
. . . . 5
⊢ (𝑗 = (𝑖 + 1) →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...(𝑖 + 1)) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 21 | 17, 20 | eqeq12d 2753 |
. . . 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)))))) |
| 22 | 21 | imbi2d 340 |
. . 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))))))) |
| 23 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑗 = 𝑁 → (𝑂‘(𝐺‘𝑗)) = (𝑂‘(𝐺‘𝑁))) |
| 24 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑗 = 𝑁 → (𝑀...𝑗) = (𝑀...𝑁)) |
| 25 | 24 | mpteq1d 5237 |
. . . . . 6
⊢ (𝑗 = 𝑁 → (𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))) |
| 26 | 25 | fveq2d 6910 |
. . . . 5
⊢ (𝑗 = 𝑁 →
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 27 | 23, 26 | eqeq12d 2753 |
. . . 4
⊢ (𝑗 = 𝑁 → ((𝑂‘(𝐺‘𝑗)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑗) ↦ (𝑂‘(𝐸‘𝑛)))) ↔ (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))) |
| 28 | 27 | imbi2d 340 |
. . 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))))))) |
| 29 | | eluzel2 12883 |
. . . . . . . . 9
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 30 | 1, 29 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 31 | | fzsn 13606 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀}) |
| 32 | 30, 31 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑀...𝑀) = {𝑀}) |
| 33 | 32 | mpteq1d 5237 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) |
| 34 | 33 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))) =
(Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))))) |
| 35 | | caratheodorylem1.o |
. . . . . . . . 9
⊢ (𝜑 → 𝑂 ∈ OutMeas) |
| 36 | 35 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑂 ∈ OutMeas) |
| 37 | | eqid 2737 |
. . . . . . . 8
⊢ ∪ dom 𝑂 = ∪ dom 𝑂 |
| 38 | | caratheodorylem1.s |
. . . . . . . . . . . 12
⊢ 𝑆 = (CaraGen‘𝑂) |
| 39 | 38 | caragenss 46519 |
. . . . . . . . . . 11
⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂) |
| 40 | 36, 39 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑆 ⊆ dom 𝑂) |
| 41 | | caratheodorylem1.e |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸:𝑍⟶𝑆) |
| 42 | 41 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝐸:𝑍⟶𝑆) |
| 43 | | elsni 4643 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ {𝑀} → 𝑛 = 𝑀) |
| 44 | 43 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 = 𝑀) |
| 45 | | uzid 12893 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
| 46 | 30, 45 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
| 47 | | caratheodorylem1.z |
. . . . . . . . . . . . . 14
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 48 | 46, 47 | eleqtrrdi 2852 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ 𝑍) |
| 49 | 48 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑀 ∈ 𝑍) |
| 50 | 44, 49 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → 𝑛 ∈ 𝑍) |
| 51 | 42, 50 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ 𝑆) |
| 52 | 40, 51 | sseldd 3984 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ∈ dom 𝑂) |
| 53 | | elssuni 4937 |
. . . . . . . . 9
⊢ ((𝐸‘𝑛) ∈ dom 𝑂 → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
| 54 | 52, 53 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
| 55 | 36, 37, 54 | omecl 46518 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ {𝑀}) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) |
| 56 | | eqid 2737 |
. . . . . . 7
⊢ (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) |
| 57 | 55, 56 | fmptd 7134 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))):{𝑀}⟶(0[,]+∞)) |
| 58 | 30, 57 | sge0sn 46394 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) = ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀)) |
| 59 | | eqidd 2738 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))) |
| 60 | 32 | iuneq1d 5019 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) = ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖)) |
| 61 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑀 → (𝐸‘𝑖) = (𝐸‘𝑀)) |
| 62 | 61 | iunxsng 5090 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ 𝑍 → ∪
𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀)) |
| 63 | 48, 62 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ {𝑀} (𝐸‘𝑖) = (𝐸‘𝑀)) |
| 64 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐸‘𝑀) = (𝐸‘𝑀)) |
| 65 | 60, 63, 64 | 3eqtrrd 2782 |
. . . . . . . . 9
⊢ (𝜑 → (𝐸‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
| 66 | 65 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
| 67 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑛 = 𝑀 → (𝐸‘𝑛) = (𝐸‘𝑀)) |
| 68 | 67 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐸‘𝑀)) |
| 69 | | caratheodorylem1.g |
. . . . . . . . . 10
⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) |
| 70 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑀 → (𝑀...𝑛) = (𝑀...𝑀)) |
| 71 | 70 | iuneq1d 5019 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑀 → ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
| 72 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ (𝑀...𝑀) ∈ V |
| 73 | | fvex 6919 |
. . . . . . . . . . . 12
⊢ (𝐸‘𝑖) ∈ V |
| 74 | 72, 73 | iunex 7993 |
. . . . . . . . . . 11
⊢ ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V |
| 75 | 74 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖) ∈ V) |
| 76 | 69, 71, 48, 75 | fvmptd3 7039 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
| 77 | 76 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐺‘𝑀) = ∪
𝑖 ∈ (𝑀...𝑀)(𝐸‘𝑖)) |
| 78 | 66, 68, 77 | 3eqtr4d 2787 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝐸‘𝑛) = (𝐺‘𝑀)) |
| 79 | 78 | fveq2d 6910 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐺‘𝑀))) |
| 80 | | snidg 4660 |
. . . . . . 7
⊢ (𝑀 ∈ 𝑍 → 𝑀 ∈ {𝑀}) |
| 81 | 48, 80 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ {𝑀}) |
| 82 | | fvexd 6921 |
. . . . . 6
⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) ∈ V) |
| 83 | 59, 79, 81, 82 | fvmptd 7023 |
. . . . 5
⊢ (𝜑 → ((𝑛 ∈ {𝑀} ↦ (𝑂‘(𝐸‘𝑛)))‘𝑀) = (𝑂‘(𝐺‘𝑀))) |
| 84 | 34, 58, 83 | 3eqtrrd 2782 |
. . . 4
⊢ (𝜑 → (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 85 | 84 | a1i 11 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (𝑂‘(𝐺‘𝑀)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑀) ↦ (𝑂‘(𝐸‘𝑛)))))) |
| 86 | | simp3 1139 |
. . . . 5
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝜑) |
| 87 | | simp1 1137 |
. . . . 5
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ (𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → 𝑖 ∈ (𝑀..^𝑁)) |
| 88 | | 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)))))) |
| 89 | 88 | imp 406 |
. . . . . 6
⊢ (((𝜑 → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) ∧ 𝜑) → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 90 | 89 | 3adant1 1131 |
. . . . 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))))) |
| 91 | | elfzoel1 13697 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℤ) |
| 92 | | elfzoelz 13699 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℤ) |
| 93 | 92 | peano2zd 12725 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℤ) |
| 94 | 91 | zred 12722 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ∈ ℝ) |
| 95 | 93 | zred 12722 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ ℝ) |
| 96 | 92 | zred 12722 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ ℝ) |
| 97 | | elfzole1 13707 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑖) |
| 98 | 96 | ltp1d 12198 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 < (𝑖 + 1)) |
| 99 | 94, 96, 95, 97, 98 | lelttrd 11419 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 < (𝑖 + 1)) |
| 100 | 94, 95, 99 | ltled 11409 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑀 ≤ (𝑖 + 1)) |
| 101 | | leid 11357 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 + 1) ∈ ℝ →
(𝑖 + 1) ≤ (𝑖 + 1)) |
| 102 | 95, 101 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ≤ (𝑖 + 1)) |
| 103 | 91, 93, 93, 100, 102 | elfzd 13555 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ (𝑀..^𝑁) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1))) |
| 104 | 103 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑀...(𝑖 + 1))) |
| 105 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑖 + 1) → (𝐸‘𝑗) = (𝐸‘(𝑖 + 1))) |
| 106 | 105 | ssiun2s 5048 |
. . . . . . . . . . . 12
⊢ ((𝑖 + 1) ∈ (𝑀...(𝑖 + 1)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
| 107 | 104, 106 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
| 108 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑗 → (𝐸‘𝑖) = (𝐸‘𝑗)) |
| 109 | 108 | cbviunv 5040 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) |
| 110 | 109 | mpteq2i 5247 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 ↦ ∪
𝑖 ∈ (𝑀...𝑛)(𝐸‘𝑖)) = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)) |
| 111 | 69, 110 | eqtri 2765 |
. . . . . . . . . . . . 13
⊢ 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗)) |
| 112 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑖 + 1) → (𝑀...𝑛) = (𝑀...(𝑖 + 1))) |
| 113 | 112 | iuneq1d 5019 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑖 + 1) → ∪ 𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
| 114 | 30 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℤ) |
| 115 | 92 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℤ) |
| 116 | 115 | peano2zd 12725 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℤ) |
| 117 | 114 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ) |
| 118 | 116 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ ℝ) |
| 119 | 115 | zred 12722 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ ℝ) |
| 120 | 97 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑖) |
| 121 | 119 | ltp1d 12198 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 < (𝑖 + 1)) |
| 122 | 117, 119,
118, 120, 121 | lelttrd 11419 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 < (𝑖 + 1)) |
| 123 | 117, 118,
122 | ltled 11409 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑀 ≤ (𝑖 + 1)) |
| 124 | 114, 116,
123 | 3jca 1129 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1))) |
| 125 | | eluz2 12884 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 + 1) ∈
(ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑖 + 1) ∈ ℤ ∧ 𝑀 ≤ (𝑖 + 1))) |
| 126 | 124, 125 | sylibr 234 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈
(ℤ≥‘𝑀)) |
| 127 | 47 | eqcomi 2746 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑀) = 𝑍 |
| 128 | 126, 127 | eleqtrdi 2851 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ 𝑍) |
| 129 | | ovex 7464 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...(𝑖 + 1)) ∈ V |
| 130 | | fvex 6919 |
. . . . . . . . . . . . . . 15
⊢ (𝐸‘𝑗) ∈ V |
| 131 | 129, 130 | iunex 7993 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V |
| 132 | 131 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ∈ V) |
| 133 | 111, 113,
128, 132 | fvmptd3 7039 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = ∪
𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗)) |
| 134 | 133 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (𝐺‘(𝑖 + 1))) |
| 135 | 107, 134 | sseqtrd 4020 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1))) |
| 136 | | sseqin2 4223 |
. . . . . . . . . . 11
⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) ↔ ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
| 137 | 136 | biimpi 216 |
. . . . . . . . . 10
⊢ ((𝐸‘(𝑖 + 1)) ⊆ (𝐺‘(𝑖 + 1)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
| 138 | 135, 137 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1))) = (𝐸‘(𝑖 + 1))) |
| 139 | 138 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐸‘(𝑖 + 1)))) |
| 140 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗(𝐸‘(𝑖 + 1)) |
| 141 | | elfzouz 13703 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ (𝑀..^𝑁) → 𝑖 ∈ (ℤ≥‘𝑀)) |
| 142 | 141 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ (ℤ≥‘𝑀)) |
| 143 | 140, 142,
105 | iunp1 45071 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) = (∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1)))) |
| 144 | 133, 143 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) = (∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1)))) |
| 145 | 144 | difeq1d 4125 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ((∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ (𝐸‘(𝑖 + 1))) ∖ (𝐸‘(𝑖 + 1)))) |
| 146 | | caratheodorylem1.dj |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → Disj 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
| 147 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (𝐸‘𝑛) = (𝐸‘𝑗)) |
| 148 | 147 | cbvdisjv 5121 |
. . . . . . . . . . . . . . 15
⊢
(Disj 𝑛
∈ 𝑍 (𝐸‘𝑛) ↔ Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
| 149 | 146, 148 | sylib 218 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
| 150 | 149 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → Disj 𝑗 ∈ 𝑍 (𝐸‘𝑗)) |
| 151 | | fzssuz 13605 |
. . . . . . . . . . . . . . 15
⊢ (𝑀...𝑖) ⊆ (ℤ≥‘𝑀) |
| 152 | 151, 127 | sseqtri 4032 |
. . . . . . . . . . . . . 14
⊢ (𝑀...𝑖) ⊆ 𝑍 |
| 153 | 152 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑀...𝑖) ⊆ 𝑍) |
| 154 | | fzp1nel 13651 |
. . . . . . . . . . . . . . . 16
⊢ ¬
(𝑖 + 1) ∈ (𝑀...𝑖) |
| 155 | 154 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ (𝑀..^𝑁) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖)) |
| 156 | 155 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ¬ (𝑖 + 1) ∈ (𝑀...𝑖)) |
| 157 | 128, 156 | eldifd 3962 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑖 + 1) ∈ (𝑍 ∖ (𝑀...𝑖))) |
| 158 | 150, 153,
157, 105 | disjiun2 45063 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∩ (𝐸‘(𝑖 + 1))) = ∅) |
| 159 | | undif4 4467 |
. . . . . . . . . . . 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)))) |
| 160 | 158, 159 | 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)))) |
| 161 | 160 | eqcomd 2743 |
. . . . . . . . . 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))))) |
| 162 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝜑) |
| 163 | 142, 127 | eleqtrdi 2851 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑖 ∈ 𝑍) |
| 164 | 111 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐺 = (𝑛 ∈ 𝑍 ↦ ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗))) |
| 165 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → 𝑛 = 𝑖) |
| 166 | 165 | oveq2d 7447 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → (𝑀...𝑛) = (𝑀...𝑖)) |
| 167 | 166 | iuneq1d 5019 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑛 = 𝑖) → ∪
𝑗 ∈ (𝑀...𝑛)(𝐸‘𝑗) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
| 168 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍) |
| 169 | | ovex 7464 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀...𝑖) ∈ V |
| 170 | 169, 130 | iunex 7993 |
. . . . . . . . . . . . . . . 16
⊢ ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V |
| 171 | 170 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∈ V) |
| 172 | 164, 167,
168, 171 | fvmptd 7023 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
| 173 | 162, 163,
172 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) = ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗)) |
| 174 | 173 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) = (𝐺‘𝑖)) |
| 175 | | difid 4376 |
. . . . . . . . . . . . 13
⊢ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅ |
| 176 | 175 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = ∅) |
| 177 | 174, 176 | uneq12d 4169 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = ((𝐺‘𝑖) ∪ ∅)) |
| 178 | | un0 4394 |
. . . . . . . . . . . 12
⊢ ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖) |
| 179 | 178 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘𝑖) ∪ ∅) = (𝐺‘𝑖)) |
| 180 | 177, 179 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ∪ ((𝐸‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝐺‘𝑖)) |
| 181 | 145, 161,
180 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))) = (𝐺‘𝑖)) |
| 182 | 181 | fveq2d 6910 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))) = (𝑂‘(𝐺‘𝑖))) |
| 183 | 139, 182 | oveq12d 7449 |
. . . . . . 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)))) |
| 184 | 183 | 3adant3 1133 |
. . . . . 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)))) |
| 185 | 35 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝑂 ∈ OutMeas) |
| 186 | 41 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → 𝐸:𝑍⟶𝑆) |
| 187 | 186, 128 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ∈ 𝑆) |
| 188 | | simpll 767 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝜑) |
| 189 | 91 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ∈ ℤ) |
| 190 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑗 ∈ ℤ) |
| 191 | 190 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ ℤ) |
| 192 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ (𝑀...(𝑖 + 1)) → 𝑀 ≤ 𝑗) |
| 193 | 192 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑀 ≤ 𝑗) |
| 194 | 189, 191,
193 | 3jca 1129 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) |
| 195 | | eluz2 12884 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 𝑀 ≤ 𝑗)) |
| 196 | 194, 195 | sylibr 234 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ (ℤ≥‘𝑀)) |
| 197 | 196, 127 | eleqtrdi 2851 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ (𝑀..^𝑁) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍) |
| 198 | 197 | adantll 714 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → 𝑗 ∈ 𝑍) |
| 199 | 35, 39 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑆 ⊆ dom 𝑂) |
| 200 | 199 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑆 ⊆ dom 𝑂) |
| 201 | 41 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ 𝑆) |
| 202 | 200, 201 | sseldd 3984 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ∈ dom 𝑂) |
| 203 | | elssuni 4937 |
. . . . . . . . . . . . . 14
⊢ ((𝐸‘𝑗) ∈ dom 𝑂 → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 204 | 202, 203 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 205 | 188, 198,
204 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑗 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 206 | 205 | ralrimiva 3146 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 207 | | iunss 5045 |
. . . . . . . . . . 11
⊢ (∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂 ↔ ∀𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 208 | 206, 207 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ∪ 𝑗 ∈ (𝑀...(𝑖 + 1))(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 209 | 133, 208 | eqsstrd 4018 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘(𝑖 + 1)) ⊆ ∪
dom 𝑂) |
| 210 | 185, 38, 37, 187, 209 | caragensplit 46515 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1))))) = (𝑂‘(𝐺‘(𝑖 + 1)))) |
| 211 | 210 | eqcomd 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘(𝑖 + 1))) = ((𝑂‘((𝐺‘(𝑖 + 1)) ∩ (𝐸‘(𝑖 + 1)))) +𝑒 (𝑂‘((𝐺‘(𝑖 + 1)) ∖ (𝐸‘(𝑖 + 1)))))) |
| 212 | 211 | 3adant3 1133 |
. . . . . 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)))))) |
| 213 | 185 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑂 ∈ OutMeas) |
| 214 | 162 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝜑) |
| 215 | | elfzuz 13560 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 216 | 215, 127 | eleqtrdi 2851 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑀...(𝑖 + 1)) → 𝑛 ∈ 𝑍) |
| 217 | 216 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → 𝑛 ∈ 𝑍) |
| 218 | 41, 199 | fssd 6753 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑂) |
| 219 | 218 | ffvelcdmda 7104 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ dom 𝑂) |
| 220 | 219, 53 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
| 221 | 214, 217,
220 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝐸‘𝑛) ⊆ ∪ dom
𝑂) |
| 222 | 213, 37, 221 | omecl 46518 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) ∧ 𝑛 ∈ (𝑀...(𝑖 + 1))) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞)) |
| 223 | | 2fveq3 6911 |
. . . . . . . . 9
⊢ (𝑛 = (𝑖 + 1) → (𝑂‘(𝐸‘𝑛)) = (𝑂‘(𝐸‘(𝑖 + 1)))) |
| 224 | 142, 222,
223 | sge0p1 46429 |
. . . . . . . 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))))) |
| 225 | 224 | 3adant3 1133 |
. . . . . . 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))))) |
| 226 | | id 22 |
. . . . . . . . . 10
⊢ ((𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) → (𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛))))) |
| 227 | 226 | eqcomd 2743 |
. . . . . . . . 9
⊢ ((𝑂‘(𝐺‘𝑖)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) →
(Σ^‘(𝑛 ∈ (𝑀...𝑖) ↦ (𝑂‘(𝐸‘𝑛)))) = (𝑂‘(𝐺‘𝑖))) |
| 228 | 227 | oveq1d 7446 |
. . . . . . . 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))))) |
| 229 | 228 | 3ad2ant3 1136 |
. . . . . . 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))))) |
| 230 | | simpl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝜑) |
| 231 | 152 | sseli 3979 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ (𝑀...𝑖) → 𝑗 ∈ 𝑍) |
| 232 | 231 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → 𝑗 ∈ 𝑍) |
| 233 | 230, 232,
204 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 234 | 233 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑖)) → (𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 235 | 234 | ralrimiva 3146 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 236 | | iunss 5045 |
. . . . . . . . . . . . 13
⊢ (∪ 𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂 ↔ ∀𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 237 | 235, 236 | sylibr 234 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ∪
𝑗 ∈ (𝑀...𝑖)(𝐸‘𝑗) ⊆ ∪ dom
𝑂) |
| 238 | 172, 237 | eqsstrd 4018 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐺‘𝑖) ⊆ ∪ dom
𝑂) |
| 239 | 162, 163,
238 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐺‘𝑖) ⊆ ∪ dom
𝑂) |
| 240 | 185, 37, 239 | omexrcl 46522 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐺‘𝑖)) ∈
ℝ*) |
| 241 | 107, 208 | sstrd 3994 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝐸‘(𝑖 + 1)) ⊆ ∪
dom 𝑂) |
| 242 | 185, 37, 241 | omexrcl 46522 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑂‘(𝐸‘(𝑖 + 1))) ∈
ℝ*) |
| 243 | 240, 242 | xaddcomd 45335 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑂‘(𝐺‘𝑖)) +𝑒 (𝑂‘(𝐸‘(𝑖 + 1)))) = ((𝑂‘(𝐸‘(𝑖 + 1))) +𝑒 (𝑂‘(𝐺‘𝑖)))) |
| 244 | 243 | 3adant3 1133 |
. . . . . . 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)))) |
| 245 | 225, 229,
244 | 3eqtrd 2781 |
. . . . . 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)))) |
| 246 | 184, 212,
245 | 3eqtr4d 2787 |
. . . . 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))))) |
| 247 | 86, 87, 90, 246 | syl3anc 1373 |
. . . 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))))) |
| 248 | 247 | 3exp 1120 |
. . 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))))))) |
| 249 | 10, 16, 22, 28, 85, 248 | fzind2 13824 |
. 2
⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛)))))) |
| 250 | 3, 4, 249 | sylc 65 |
1
⊢ (𝜑 → (𝑂‘(𝐺‘𝑁)) =
(Σ^‘(𝑛 ∈ (𝑀...𝑁) ↦ (𝑂‘(𝐸‘𝑛))))) |