| Step | Hyp | Ref
| Expression |
| 1 | | itgeqa.3 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 2 | | itgeqa.4 |
. . . . 5
⊢ (𝜑 → (vol*‘𝐴) = 0) |
| 3 | | itgeqa.5 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷) |
| 4 | | itgeqa.1 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 5 | | itgeqa.2 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ) |
| 6 | 1, 2, 3, 4, 5 | mbfeqa 25678 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn)) |
| 7 | | ifan 4579 |
. . . . . . . . . 10
⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) |
| 8 | 4 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 9 | | ax-icn 11214 |
. . . . . . . . . . . . . . . . 17
⊢ i ∈
ℂ |
| 10 | | ine0 11698 |
. . . . . . . . . . . . . . . . 17
⊢ i ≠
0 |
| 11 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...3) → 𝑘 ∈
ℤ) |
| 12 | 11 | ad2antlr 727 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℤ) |
| 13 | | expclz 14125 |
. . . . . . . . . . . . . . . . 17
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈
ℂ) |
| 14 | 9, 10, 12, 13 | mp3an12i 1467 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ∈ ℂ) |
| 15 | | expne0i 14135 |
. . . . . . . . . . . . . . . . 17
⊢ ((i
∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0) |
| 16 | 9, 10, 12, 15 | mp3an12i 1467 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ≠ 0) |
| 17 | 8, 14, 16 | divcld 12043 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ) |
| 18 | 17 | recld 15233 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) |
| 19 | | 0re 11263 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
| 20 | | ifcl 4571 |
. . . . . . . . . . . . . 14
⊢
(((ℜ‘(𝐶 /
(i↑𝑘))) ∈ ℝ
∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ) |
| 21 | 18, 19, 20 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ) |
| 22 | 21 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
ℝ*) |
| 23 | | max1 13227 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
| 24 | 19, 18, 23 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤
(ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
| 25 | | elxrge0 13497 |
. . . . . . . . . . . 12
⊢ (if(0
≤ (ℜ‘(𝐶 /
(i↑𝑘))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0) ∈
(0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ* ∧ 0
≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
| 26 | 22, 24, 25 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 27 | | 0e0iccpnf 13499 |
. . . . . . . . . . . 12
⊢ 0 ∈
(0[,]+∞) |
| 28 | 27 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ∈
(0[,]+∞)) |
| 29 | 26, 28 | ifclda 4561 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈
(0[,]+∞)) |
| 30 | 7, 29 | eqeltrid 2845 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 31 | 30 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 32 | 31 | fmpttd 7135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))),
0)):ℝ⟶(0[,]+∞)) |
| 33 | | ifan 4579 |
. . . . . . . . . 10
⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) = if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) |
| 34 | 5 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ) |
| 35 | 34, 14, 16 | divcld 12043 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ) |
| 36 | 35 | recld 15233 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) |
| 37 | | ifcl 4571 |
. . . . . . . . . . . . . 14
⊢
(((ℜ‘(𝐷 /
(i↑𝑘))) ∈ ℝ
∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ) |
| 38 | 36, 19, 37 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ) |
| 39 | 38 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈
ℝ*) |
| 40 | | max1 13227 |
. . . . . . . . . . . . 13
⊢ ((0
∈ ℝ ∧ (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤
(ℜ‘(𝐷 /
(i↑𝑘))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0)) |
| 41 | 19, 36, 40 | sylancr 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤
(ℜ‘(𝐷 /
(i↑𝑘))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0)) |
| 42 | | elxrge0 13497 |
. . . . . . . . . . . 12
⊢ (if(0
≤ (ℜ‘(𝐷 /
(i↑𝑘))),
(ℜ‘(𝐷 /
(i↑𝑘))), 0) ∈
(0[,]+∞) ↔ (if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ* ∧ 0
≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) |
| 43 | 39, 41, 42 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 44 | 43, 28 | ifclda 4561 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈
(0[,]+∞)) |
| 45 | 33, 44 | eqeltrid 2845 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 46 | 45 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈
(0[,]+∞)) |
| 47 | 46 | fmpttd 7135 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))),
0)):ℝ⟶(0[,]+∞)) |
| 48 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ) |
| 49 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0) |
| 50 | | simpll 767 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝜑) |
| 51 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) |
| 52 | | eldifn 4132 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴) |
| 53 | 52 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴) |
| 54 | 51, 53 | eldifd 3962 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∖ 𝐴)) |
| 55 | 50, 54, 3 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
| 56 | 55 | fvoveq1d 7453 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))) |
| 57 | 56 | ibllem 25799 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) |
| 58 | | eldifi 4131 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ) |
| 59 | 58 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ) |
| 60 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢
(ℜ‘(𝐶 /
(i↑𝑘))) ∈
V |
| 61 | | c0ex 11255 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
| 62 | 60, 61 | ifex 4576 |
. . . . . . . . . . . . 13
⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V |
| 63 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)) |
| 64 | 63 | fvmpt2 7027 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
| 65 | 59, 62, 64 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) |
| 66 | | fvex 6919 |
. . . . . . . . . . . . . 14
⊢
(ℜ‘(𝐷 /
(i↑𝑘))) ∈
V |
| 67 | 66, 61 | ifex 4576 |
. . . . . . . . . . . . 13
⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V |
| 68 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0)) |
| 69 | 68 | fvmpt2 7027 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) |
| 70 | 59, 67, 69 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) |
| 71 | 57, 65, 70 | 3eqtr4d 2787 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥)) |
| 72 | 71 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥)) |
| 73 | | nfv 1914 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))‘𝑥) = ((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0))‘𝑥) |
| 74 | | nffvmpt1 6917 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) |
| 75 | | nffvmpt1 6917 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦) |
| 76 | 74, 75 | nfeq 2919 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))‘𝑦) = ((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0))‘𝑦) |
| 77 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)) |
| 78 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)) |
| 79 | 77, 78 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) ↔ ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))) |
| 80 | 73, 76, 79 | cbvralw 3306 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
(ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))‘𝑥) = ((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0))‘𝑥) ↔
∀𝑦 ∈ (ℝ
∖ 𝐴)((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0))‘𝑦) = ((𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0))‘𝑦)) |
| 81 | 72, 80 | sylib 218 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑦 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)) |
| 82 | 81 | r19.21bi 3251 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)) |
| 83 | 82 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑦 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)) |
| 84 | 32, 47, 48, 49, 83 | itg2eqa 25780 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0)))) |
| 85 | 84 | eleq1d 2826 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) →
((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ↔
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ)) |
| 86 | 85 | ralbidva 3176 |
. . . 4
⊢ (𝜑 → (∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ)) |
| 87 | 6, 86 | anbi12d 632 |
. . 3
⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) ↔ ((𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ))) |
| 88 | | eqidd 2738 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) |
| 89 | | eqidd 2738 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))) |
| 90 | 88, 89, 4 | isibl2 25801 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ))) |
| 91 | | eqidd 2738 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) |
| 92 | | eqidd 2738 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))) |
| 93 | 91, 92, 5 | isibl2 25801 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1 ↔
((𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn ∧ ∀𝑘 ∈
(0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ))) |
| 94 | 87, 90, 93 | 3bitr4d 311 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈
𝐿1)) |
| 95 | 84 | oveq2d 7447 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → ((i↑𝑘) ·
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0))))) |
| 96 | 95 | sumeq2dv 15738 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))), 0)))) =
Σ𝑘 ∈
(0...3)((i↑𝑘) ·
(∫2‘(𝑥
∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))) |
| 97 | | eqid 2737 |
. . . 4
⊢
(ℜ‘(𝐶 /
(i↑𝑘))) =
(ℜ‘(𝐶 /
(i↑𝑘))) |
| 98 | 97 | dfitg 25804 |
. . 3
⊢
∫𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐶 /
(i↑𝑘)))),
(ℜ‘(𝐶 /
(i↑𝑘))),
0)))) |
| 99 | | eqid 2737 |
. . . 4
⊢
(ℜ‘(𝐷 /
(i↑𝑘))) =
(ℜ‘(𝐷 /
(i↑𝑘))) |
| 100 | 99 | dfitg 25804 |
. . 3
⊢
∫𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦
if((𝑥 ∈ 𝐵 ∧ 0 ≤
(ℜ‘(𝐷 /
(i↑𝑘)))),
(ℜ‘(𝐷 /
(i↑𝑘))),
0)))) |
| 101 | 96, 98, 100 | 3eqtr4g 2802 |
. 2
⊢ (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥) |
| 102 | 94, 101 | jca 511 |
1
⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1) ∧
∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)) |