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Theorem itgeqa 24406
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 24236 . . . 4 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))
7 ifan 4516 . . . . . . . . . 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 713 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
9 ax-icn 10588 . . . . . . . . . . . . . . . . 17 i ∈ ℂ
10 ine0 11067 . . . . . . . . . . . . . . . . 17 i ≠ 0
11 elfzelz 12900 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
1211ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
13 expclz 13446 . . . . . . . . . . . . . . . . 17 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
149, 10, 12, 13mp3an12i 1458 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
15 expne0i 13453 . . . . . . . . . . . . . . . . 17 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
169, 10, 12, 15mp3an12i 1458 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
178, 14, 16divcld 11408 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
1817recld 14545 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
19 0re 10635 . . . . . . . . . . . . . 14 0 ∈ ℝ
20 ifcl 4509 . . . . . . . . . . . . . 14 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2118, 19, 20sylancl 588 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2221rexrd 10683 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
23 max1 12570 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
2419, 18, 23sylancr 589 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
25 elxrge0 12837 . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
27 0e0iccpnf 12839 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
2827a1i 11 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
2926, 28ifclda 4499 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
307, 29eqeltrid 2915 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3130adantr 483 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3231fmpttd 6872 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
33 ifan 4516 . . . . . . . . . 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 713 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐷 ∈ ℂ)
3534, 14, 16divcld 11408 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
3635recld 14545 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ)
37 ifcl 4509 . . . . . . . . . . . . . 14 (((ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
3836, 19, 37sylancl 588 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
3938rexrd 10683 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ*)
40 max1 12570 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
4119, 36, 40sylancr 589 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
42 elxrge0 12837 . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4443, 28ifclda 4499 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4533, 44eqeltrid 2915 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4645adantr 483 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4746fmpttd 6872 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
481adantr 483 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
492adantr 483 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
50 simpll 765 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝜑)
51 simpr 487 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥𝐵)
52 eldifn 4102 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
5352ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → ¬ 𝑥𝐴)
5451, 53eldifd 3945 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐵𝐴))
5550, 54, 3syl2anc 586 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝐶 = 𝐷)
5655fvoveq1d 7170 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
5756ibllem 24357 . . . . . . . . . . . 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 4101 . . . . . . . . . . . . . 14 (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
5958adantl 484 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
60 fvex 6676 . . . . . . . . . . . . . 14 (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
61 c0ex 10627 . . . . . . . . . . . . . 14 0 ∈ V
6260, 61ifex 4513 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
63 eqid 2819 . . . . . . . . . . . . . 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 6772 . . . . . . . . . . . . 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 588 . . . . . . . . . . . 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 6676 . . . . . . . . . . . . . 14 (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
6766, 61ifex 4513 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
68 eqid 2819 . . . . . . . . . . . . . 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 6772 . . . . . . . . . . . . 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 588 . . . . . . . . . . . 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 2864 . . . . . . . . . . 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 3180 . . . . . . . . . 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 1908 . . . . . . . . . . 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 6674 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
75 nffvmpt1 6674 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
7674, 75nfeq 2989 . . . . . . . . . . 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 6663 . . . . . . . . . . . 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 6663 . . . . . . . . . . . 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 2835 . . . . . . . . . . 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 3440 . . . . . . . . . 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 220 . . . . . . . . 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 3206 . . . . . . . 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 713 . . . . . . 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 24338 . . . . . 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 2895 . . . . 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 3194 . . . 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 2820 . . . 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 2820 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
9088, 89, 4isibl2 24359 . . 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 2820 . . . 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 2820 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
9391, 92, 5isibl2 24359 . . 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 313 . 2 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1))
9584oveq2d 7164 . . . 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 15052 . . 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 2819 . . . 4 (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
9897dfitg 24362 . . 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 2819 . . . 4 (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
10099dfitg 24362 . . 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 2879 . 2 (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
10294, 101jca 514 1 (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wne 3014  wral 3136  Vcvv 3493  cdif 3931  wss 3934  ifcif 4465   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7148  cc 10527  cr 10528  0cc0 10529  ici 10531   · cmul 10534  +∞cpnf 10664  *cxr 10666  cle 10668   / cdiv 11289  3c3 11685  cz 11973  [,]cicc 12733  ...cfz 12884  cexp 13421  cre 14448  Σcsu 15034  vol*covol 24055  MblFncmbf 24207  2citg2 24209  𝐿1cibl 24210  citg 24211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-symdif 4217  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-ofr 7402  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057  df-vol 24058  df-mbf 24212  df-itg1 24213  df-itg2 24214  df-ibl 24215  df-itg 24216
This theorem is referenced by:  itgss3  24407
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