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Theorem itgeqa 23887
Description: Approximate equality of integrals. If 𝐶(𝑥) = 𝐷(𝑥) for almost all 𝑥, then 𝐵𝐶(𝑥) d𝑥 = ∫𝐵𝐷(𝑥) d𝑥 and one is integrable iff the other is. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 2-Sep-2014.)
Hypotheses
Ref Expression
itgeqa.1 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
itgeqa.2 ((𝜑𝑥𝐵) → 𝐷 ∈ ℂ)
itgeqa.3 (𝜑𝐴 ⊆ ℝ)
itgeqa.4 (𝜑 → (vol*‘𝐴) = 0)
itgeqa.5 ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)
Assertion
Ref Expression
itgeqa (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem itgeqa
Dummy variables 𝑦 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 itgeqa.3 . . . . 5 (𝜑𝐴 ⊆ ℝ)
2 itgeqa.4 . . . . 5 (𝜑 → (vol*‘𝐴) = 0)
3 itgeqa.5 . . . . 5 ((𝜑𝑥 ∈ (𝐵𝐴)) → 𝐶 = 𝐷)
4 itgeqa.1 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 ∈ ℂ)
5 itgeqa.2 . . . . 5 ((𝜑𝑥𝐵) → 𝐷 ∈ ℂ)
61, 2, 3, 4, 5mbfeqa 23717 . . . 4 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))
7 ifan 4296 . . . . . . . . . 10 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0)
84adantlr 706 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
9 elfzelz 12556 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
109ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
11 ax-icn 10252 . . . . . . . . . . . . . . . . . 18 i ∈ ℂ
12 ine0 10724 . . . . . . . . . . . . . . . . . 18 i ≠ 0
13 expclz 13099 . . . . . . . . . . . . . . . . . 18 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
1411, 12, 13mp3an12 1575 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
1510, 14syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
16 expne0i 13106 . . . . . . . . . . . . . . . . . 18 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
1711, 12, 16mp3an12 1575 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
1810, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
198, 15, 18divcld 11060 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
2019recld 14235 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
21 0re 10299 . . . . . . . . . . . . . 14 0 ∈ ℝ
22 ifcl 4289 . . . . . . . . . . . . . 14 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2320, 21, 22sylancl 580 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2423rexrd 10347 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
25 max1 12225 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
2621, 20, 25sylancr 581 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
27 elxrge0 12492 . . . . . . . . . . . 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)))
2824, 26, 27sylanbrc 578 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
29 0e0iccpnf 12494 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
3029a1i 11 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
3128, 30ifclda 4279 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
327, 31syl5eqel 2848 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3332adantr 472 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3433fmpttd 6579 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
35 ifan 4296 . . . . . . . . . 10 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) = if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0)
365adantlr 706 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐷 ∈ ℂ)
3736, 15, 18divcld 11060 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
3837recld 14235 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ)
39 ifcl 4289 . . . . . . . . . . . . . 14 (((ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
4038, 21, 39sylancl 580 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
4140rexrd 10347 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ*)
42 max1 12225 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
4321, 38, 42sylancr 581 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
44 elxrge0 12492 . . . . . . . . . . . 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)))
4541, 43, 44sylanbrc 578 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4645, 30ifclda 4279 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4735, 46syl5eqel 2848 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4847adantr 472 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4948fmpttd 6579 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
501adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
512adantr 472 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
52 simpll 783 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝜑)
53 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥𝐵)
54 eldifn 3897 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
5554ad2antlr 718 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → ¬ 𝑥𝐴)
5653, 55eldifd 3745 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐵𝐴))
5752, 56, 3syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝐶 = 𝐷)
5857fvoveq1d 6868 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
5958ibllem 23838 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
60 eldifi 3896 . . . . . . . . . . . . . 14 (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
6160adantl 473 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
62 fvex 6392 . . . . . . . . . . . . . 14 (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
63 c0ex 10291 . . . . . . . . . . . . . 14 0 ∈ V
6462, 63ifex 4293 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
65 eqid 2765 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
6665fvmpt2 6484 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
6761, 64, 66sylancl 580 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
68 fvex 6392 . . . . . . . . . . . . . 14 (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
6968, 63ifex 4293 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
70 eqid 2765 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
7170fvmpt2 6484 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
7261, 69, 71sylancl 580 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
7359, 67, 723eqtr4d 2809 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥))
7473ralrimiva 3113 . . . . . . . . . 10 (𝜑 → ∀𝑥 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥))
75 nfv 2009 . . . . . . . . . . 11 𝑦((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥)
76 nffvmpt1 6390 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
77 nffvmpt1 6390 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
7876, 77nfeq 2919 . . . . . . . . . . 11 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
79 fveq2 6379 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦))
80 fveq2 6379 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))
8179, 80eqeq12d 2780 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) ↔ ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)))
8275, 78, 81cbvral 3315 . . . . . . . . . 10 (∀𝑥 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑥) ↔ ∀𝑦 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))
8374, 82sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (ℝ ∖ 𝐴)((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))
8483r19.21bi 3079 . . . . . . . 8 ((𝜑𝑦 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))
8584adantlr 706 . . . . . . 7 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑦 ∈ (ℝ ∖ 𝐴)) → ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦))
8634, 49, 50, 51, 85itg2eqa 23819 . . . . . 6 ((𝜑𝑘 ∈ (0...3)) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))
8786eleq1d 2829 . . . . 5 ((𝜑𝑘 ∈ (0...3)) → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ))
8887ralbidva 3132 . . . 4 (𝜑 → (∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ ↔ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ))
896, 88anbi12d 624 . . 3 (𝜑 → (((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ) ↔ ((𝑥𝐵𝐷) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ)))
90 eqidd 2766 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))
91 eqidd 2766 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
9290, 91, 4isibl2 23840 . . 3 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ ((𝑥𝐵𝐶) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))) ∈ ℝ)))
93 eqidd 2766 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)))
94 eqidd 2766 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
9593, 94, 5isibl2 23840 . . 3 (𝜑 → ((𝑥𝐵𝐷) ∈ 𝐿1 ↔ ((𝑥𝐵𝐷) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))) ∈ ℝ)))
9689, 92, 953bitr4d 302 . 2 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1))
9786oveq2d 6862 . . . 4 ((𝜑𝑘 ∈ (0...3)) → ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)))))
9897sumeq2dv 14734 . . 3 (𝜑 → Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)))) = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)))))
99 eqid 2765 . . . 4 (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
10099dfitg 23843 . . 3 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
101 eqid 2765 . . . 4 (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
102101dfitg 23843 . . 3 𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))
10398, 100, 1023eqtr4g 2824 . 2 (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
10496, 103jca 507 1 (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3731  wss 3734  ifcif 4245   class class class wbr 4811  cmpt 4890  cfv 6070  (class class class)co 6846  cc 10191  cr 10192  0cc0 10193  ici 10195   · cmul 10198  +∞cpnf 10329  *cxr 10331  cle 10333   / cdiv 10943  3c3 11333  cz 11629  [,]cicc 12387  ...cfz 12540  cexp 13074  cre 14138  Σcsu 14717  vol*covol 23536  MblFncmbf 23688  2citg2 23690  𝐿1cibl 23691  citg 23692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757  ax-cnex 10249  ax-resscn 10250  ax-1cn 10251  ax-icn 10252  ax-addcl 10253  ax-addrcl 10254  ax-mulcl 10255  ax-mulrcl 10256  ax-mulcom 10257  ax-addass 10258  ax-mulass 10259  ax-distr 10260  ax-i2m1 10261  ax-1ne0 10262  ax-1rid 10263  ax-rnegex 10264  ax-rrecex 10265  ax-cnre 10266  ax-pre-lttri 10267  ax-pre-lttrn 10268  ax-pre-ltadd 10269  ax-pre-mulgt0 10270  ax-pre-sup 10271  ax-addf 10272
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-symdif 4007  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-disj 4780  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6807  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-of 7099  df-ofr 7100  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-2o 7769  df-oadd 7772  df-er 7951  df-map 8066  df-pm 8067  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168  df-fi 8528  df-sup 8559  df-inf 8560  df-oi 8626  df-card 9020  df-cda 9247  df-pnf 10334  df-mnf 10335  df-xr 10336  df-ltxr 10337  df-le 10338  df-sub 10527  df-neg 10528  df-div 10944  df-nn 11280  df-2 11340  df-3 11341  df-n0 11544  df-z 11630  df-uz 11894  df-q 11997  df-rp 12036  df-xneg 12153  df-xadd 12154  df-xmul 12155  df-ioo 12388  df-ico 12390  df-icc 12391  df-fz 12541  df-fzo 12681  df-fl 12808  df-seq 13016  df-exp 13075  df-hash 13329  df-cj 14140  df-re 14141  df-im 14142  df-sqrt 14276  df-abs 14277  df-clim 14520  df-sum 14718  df-rest 16365  df-topgen 16386  df-psmet 20027  df-xmet 20028  df-met 20029  df-bl 20030  df-mopn 20031  df-top 20994  df-topon 21011  df-bases 21046  df-cmp 21486  df-ovol 23538  df-vol 23539  df-mbf 23693  df-itg1 23694  df-itg2 23695  df-ibl 23696  df-itg 23697
This theorem is referenced by:  itgss3  23888
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