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Theorem itgeqa 25849
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 25678 . . . 4 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))
7 ifan 4579 . . . . . . . . . 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 715 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
9 ax-icn 11214 . . . . . . . . . . . . . . . . 17 i ∈ ℂ
10 ine0 11698 . . . . . . . . . . . . . . . . 17 i ≠ 0
11 elfzelz 13564 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
1211ad2antlr 727 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
13 expclz 14125 . . . . . . . . . . . . . . . . 17 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
149, 10, 12, 13mp3an12i 1467 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
15 expne0i 14135 . . . . . . . . . . . . . . . . 17 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
169, 10, 12, 15mp3an12i 1467 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
178, 14, 16divcld 12043 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
1817recld 15233 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (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)
2118, 19, 20sylancl 586 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2221rexrd 11311 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 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))
2419, 18, 23sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 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)))
2622, 24, 25sylanbrc 583 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
27 0e0iccpnf 13499 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
2827a1i 11 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
2926, 28ifclda 4561 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
307, 29eqeltrid 2845 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3130adantr 480 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3231fmpttd 7135 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
33 ifan 4579 . . . . . . . . . 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)
345adantlr 715 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐷 ∈ ℂ)
3534, 14, 16divcld 12043 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
3635recld 15233 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (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)
3836, 19, 37sylancl 586 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
3938rexrd 11311 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 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))
4119, 36, 40sylancr 587 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 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)))
4339, 41, 42sylanbrc 583 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4443, 28ifclda 4561 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4533, 44eqeltrid 2845 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4645adantr 480 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4746fmpttd 7135 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
481adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
492adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
50 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝜑)
51 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥𝐵)
52 eldifn 4132 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
5352ad2antlr 727 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → ¬ 𝑥𝐴)
5451, 53eldifd 3962 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐵𝐴))
5550, 54, 3syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝐶 = 𝐷)
5655fvoveq1d 7453 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
5756ibllem 25799 . . . . . . . . . . . 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))
58 eldifi 4131 . . . . . . . . . . . . . 14 (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
5958adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
60 fvex 6919 . . . . . . . . . . . . . 14 (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
61 c0ex 11255 . . . . . . . . . . . . . 14 0 ∈ V
6260, 61ifex 4576 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
63 eqid 2737 . . . . . . . . . . . . . 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))
6463fvmpt2 7027 . . . . . . . . . . . . 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))
6559, 62, 64sylancl 586 . . . . . . . . . . . 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))
66 fvex 6919 . . . . . . . . . . . . . 14 (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
6766, 61ifex 4576 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
68 eqid 2737 . . . . . . . . . . . . . 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))
6968fvmpt2 7027 . . . . . . . . . . . . 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))
7059, 67, 69sylancl 586 . . . . . . . . . . . 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))
7157, 65, 703eqtr4d 2787 . . . . . . . . . . 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))
7271ralrimiva 3146 . . . . . . . . . 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))
73 nfv 1914 . . . . . . . . . . 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)
74 nffvmpt1 6917 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
75 nffvmpt1 6917 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
7674, 75nfeq 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)
77 fveq2 6906 . . . . . . . . . . . 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))
78 fveq2 6906 . . . . . . . . . . . 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))
7977, 78eqeq12d 2753 . . . . . . . . . . 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)))
8073, 76, 79cbvralw 3306 . . . . . . . . . 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))
8172, 80sylib 218 . . . . . . . . 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))
8281r19.21bi 3251 . . . . . . . 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))
8382adantlr 715 . . . . . . 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))
8432, 47, 48, 49, 83itg2eqa 25780 . . . . . 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))))
8584eleq1d 2826 . . . . 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))
8685ralbidva 3176 . . . 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))
876, 86anbi12d 632 . . 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)))
88 eqidd 2738 . . . 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)))
89 eqidd 2738 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
9088, 89, 4isibl2 25801 . . 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)))
91 eqidd 2738 . . . 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)))
92 eqidd 2738 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
9391, 92, 5isibl2 25801 . . 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)))
9487, 90, 933bitr4d 311 . 2 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1))
9584oveq2d 7447 . . . 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)))))
9695sumeq2dv 15738 . . 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)))))
97 eqid 2737 . . . 4 (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
9897dfitg 25804 . . 3 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
99 eqid 2737 . . . 4 (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
10099dfitg 25804 . . 3 𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))
10196, 98, 1003eqtr4g 2802 . 2 (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
10294, 101jca 511 1 (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cdif 3948  wss 3951  ifcif 4525   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  ici 11157   · cmul 11160  +∞cpnf 11292  *cxr 11294  cle 11296   / cdiv 11920  3c3 12322  cz 12613  [,]cicc 13390  ...cfz 13547  cexp 14102  cre 15136  Σcsu 15722  vol*covol 25497  MblFncmbf 25649  2citg2 25651  𝐿1cibl 25652  citg 25653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-symdif 4253  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656  df-ibl 25657  df-itg 25658
This theorem is referenced by:  itgss3  25850
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