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Theorem itgeqa 23386
 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 23216 . . . 4 (𝜑 → ((𝑥𝐵𝐶) ∈ MblFn ↔ (𝑥𝐵𝐷) ∈ MblFn))
7 ifan 4084 . . . . . . . . . 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 747 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐶 ∈ ℂ)
9 elfzelz 12213 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
109ad2antlr 759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝑘 ∈ ℤ)
11 ax-icn 9874 . . . . . . . . . . . . . . . . . 18 i ∈ ℂ
12 ine0 10344 . . . . . . . . . . . . . . . . . 18 i ≠ 0
13 expclz 12747 . . . . . . . . . . . . . . . . . 18 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
1411, 12, 13mp3an12 1406 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
1510, 14syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ∈ ℂ)
16 expne0i 12754 . . . . . . . . . . . . . . . . . 18 ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
1711, 12, 16mp3an12 1406 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
1810, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (i↑𝑘) ≠ 0)
198, 15, 18divcld 10680 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
2019recld 13782 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
21 0re 9919 . . . . . . . . . . . . . 14 0 ∈ ℝ
22 ifcl 4080 . . . . . . . . . . . . . 14 (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2320, 21, 22sylancl 693 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
2423rexrd 9968 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
25 max1 11890 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
2621, 20, 25sylancr 694 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
27 elxrge0 12152 . . . . . . . . . . . 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 695 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
29 0e0iccpnf 12154 . . . . . . . . . . . 12 0 ∈ (0[,]+∞)
3029a1i 11 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ ¬ 𝑥𝐵) → 0 ∈ (0[,]+∞))
3128, 30ifclda 4070 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
327, 31syl5eqel 2692 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
3332adantr 480 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
34 eqid 2610 . . . . . . . 8 (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))
3533, 34fmptd 6292 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
36 ifan 4084 . . . . . . . . . 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)
375adantlr 747 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 𝐷 ∈ ℂ)
3837, 15, 18divcld 10680 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
3938recld 13782 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ)
40 ifcl 4080 . . . . . . . . . . . . . 14 (((ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
4139, 21, 40sylancl 693 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
4241rexrd 9968 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ*)
43 max1 11890 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
4421, 39, 43sylancr 694 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
45 elxrge0 12152 . . . . . . . . . . . 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)))
4642, 44, 45sylanbrc 695 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4746, 30ifclda 4070 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...3)) → if(𝑥𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
4836, 47syl5eqel 2692 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...3)) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
4948adantr 480 . . . . . . . 8 (((𝜑𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
50 eqid 2610 . . . . . . . 8 (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))
5149, 50fmptd 6292 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
521adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
532adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
54 simpll 786 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝜑)
55 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥𝐵)
56 eldifn 3695 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥𝐴)
5756ad2antlr 759 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → ¬ 𝑥𝐴)
5855, 57eldifd 3551 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐵𝐴))
5954, 58, 3syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → 𝐶 = 𝐷)
6059oveq1d 6564 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘)))
6160fveq2d 6107 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
6261ibllem 23337 . . . . . . . . . . . 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))
63 eldifi 3694 . . . . . . . . . . . . . 14 (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
6463adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
65 fvex 6113 . . . . . . . . . . . . . 14 (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
66 c0ex 9913 . . . . . . . . . . . . . 14 0 ∈ V
6765, 66ifex 4106 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
6834fvmpt2 6200 . . . . . . . . . . . . 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))
6964, 67, 68sylancl 693 . . . . . . . . . . . 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))
70 fvex 6113 . . . . . . . . . . . . . 14 (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
7170, 66ifex 4106 . . . . . . . . . . . . 13 if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
7250fvmpt2 6200 . . . . . . . . . . . . 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))
7364, 71, 72sylancl 693 . . . . . . . . . . . 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))
7462, 69, 733eqtr4d 2654 . . . . . . . . . . 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))
7574ralrimiva 2949 . . . . . . . . . 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))
76 nfv 1830 . . . . . . . . . . 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)
77 nffvmpt1 6111 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
78 nffvmpt1 6111 . . . . . . . . . . . 12 𝑥((𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
7977, 78nfeq 2762 . . . . . . . . . . 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)
80 fveq2 6103 . . . . . . . . . . . 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))
81 fveq2 6103 . . . . . . . . . . . 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))
8280, 81eqeq12d 2625 . . . . . . . . . . 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)))
8376, 79, 82cbvral 3143 . . . . . . . . . 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))
8475, 83sylib 207 . . . . . . . . 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))
8584r19.21bi 2916 . . . . . . . 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))
8685adantlr 747 . . . . . . 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))
8735, 51, 52, 53, 86itg2eqa 23318 . . . . . 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))))
8887eleq1d 2672 . . . . 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))
8988ralbidva 2968 . . . 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))
906, 89anbi12d 743 . . 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)))
91 eqidd 2611 . . . 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)))
92 eqidd 2611 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
9391, 92, 4isibl2 23339 . . 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)))
94 eqidd 2611 . . . 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)))
95 eqidd 2611 . . . 4 ((𝜑𝑥𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
9694, 95, 5isibl2 23339 . . 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)))
9790, 93, 963bitr4d 299 . 2 (𝜑 → ((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1))
9887oveq2d 6565 . . . 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)))))
9998sumeq2dv 14281 . . 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)))))
100 eqid 2610 . . . 4 (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
101100dfitg 23342 . . 3 𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
102 eqid 2610 . . . 4 (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
103102dfitg 23342 . . 3 𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))
10499, 101, 1033eqtr4g 2669 . 2 (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
10597, 104jca 553 1 (𝜑 → (((𝑥𝐵𝐶) ∈ 𝐿1 ↔ (𝑥𝐵𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  ici 9817   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   ≤ cle 9954   / cdiv 10563  3c3 10948  ℤcz 11254  [,]cicc 12049  ...cfz 12197  ↑cexp 12722  ℜcre 13685  Σcsu 14264  vol*covol 23038  MblFncmbf 23189  ∫2citg2 23191  𝐿1cibl 23192  ∫citg 23193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195  df-itg2 23196  df-ibl 23197  df-itg 23198 This theorem is referenced by:  itgss3  23387
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