Theorem itgeqa 25845

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	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
itgeqa.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ)
itgeqa.3	⊢ (𝜑 → 𝐴 ⊆ ℝ)
itgeqa.4	⊢ (𝜑 → (vol*‘𝐴) = 0)
itgeqa.5	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 𝐷)

Assertion

Ref	Expression
itgeqa	⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 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.

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 25674	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn))
7		ifan 4524	. . . . . . . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
9		ax-icn 11118	. . . . . . . . . . . . . . . . 17 ⊢ i ∈ ℂ
10		ine0 11608	. . . . . . . . . . . . . . . . 17 ⊢ i ≠ 0
11		elfzelz 13515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
12	11	ad2antlr 735	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℤ)
13		expclz 14083	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
14	9, 10, 12, 13	mp3an12i 1476	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ∈ ℂ)
15		expne0i 14093	. . . . . . . . . . . . . . . . 17 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
16	9, 10, 12, 15	mp3an12i 1476	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ≠ 0)
17	8, 14, 16	divcld 11953	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
18	17	recld 15193	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
19		0re 11169	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
20		ifcl 4516	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
22	21	rexrd 11218	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
23		max1 13174	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
25		elxrge0 13447	. . . . . . . . . . . 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 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
27		0e0iccpnf 13449	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
28	27	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ∈ (0[,]+∞))
29	26, 28	ifclda 4506	. . . . . . . . . 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 2856	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
31	30	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
32	31	fmpttd 7081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
33		ifan 4524	. . . . . . . . . 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 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ)
35	34, 14, 16	divcld 11953	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
36	35	recld 15193	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ)
37		ifcl 4516	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
39	38	rexrd 11218	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ*)
40		max1 13174	. . . . . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
42		elxrge0 13447	. . . . . . . . . . . 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 591	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
44	43, 28	ifclda 4506	. . . . . . . . . 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 2856	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
46	45	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
47	46	fmpttd 7081	. . . . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
49	2	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
50		simpll 774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝜑)
51		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
52		eldifn 4076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
53	52	ad2antlr 735	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴)
54	51, 53	eldifd 3906	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∖ 𝐴))
55	50, 54, 3	syl2anc 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
56	55	fvoveq1d 7403	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
57	56	ibllem 25795	. . . . . . . . . . . 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 4075	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
59	58	adantl 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
60		fvex 6865	. . . . . . . . . . . . . 14 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
61		c0ex 11159	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
62	60, 61	ifex 4521	. . . . . . . . . . . . 13 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
63		eqid 2752	. . . . . . . . . . . . . 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 6972	. . . . . . . . . . . . 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 594	. . . . . . . . . . . 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 6865	. . . . . . . . . . . . . 14 ⊢ (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
67	66, 61	ifex 4521	. . . . . . . . . . . . 13 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
68		eqid 2752	. . . . . . . . . . . . . 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 6972	. . . . . . . . . . . . 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 594	. . . . . . . . . . . 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 2797	. . . . . . . . . . 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 3144	. . . . . . . . . 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 1924	. . . . . . . . . . 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 6863	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
75		nffvmpt1 6863	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
76	74, 75	nfeq 2927	. . . . . . . . . . 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 6852	. . . . . . . . . . . 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 6852	. . . . . . . . . . . 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 2768	. . . . . . . . . . 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 3294	. . . . . . . . . 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 220	. . . . . . . . 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 3244	. . . . . . . 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 723	. . . . . . 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 25776	. . . . . 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 2837	. . . . 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 3173	. . . 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 640	. . 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 2753	. . . 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 2753	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
90	88, 89, 4	isibl2 25797	. . 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 2753	. . . 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 2753	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
93	91, 92, 5	isibl2 25797	. . 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 313	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1))
95	84	oveq2d 7397	. . . 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 15701	. . 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 2752	. . . 4 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
98	97	dfitg 25800	. . 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 2752	. . . 4 ⊢ (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
100	99	dfitg 25800	. . 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 2812	. 2 ⊢ (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
102	94, 101	jca 518	1 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∀wral 3066  Vcvv 3444   ∖ cdif 3892   ⊆ wss 3895  ifcif 4470   class class class wbr 5090   ↦ cmpt 5171  ‘cfv 6506  (class class class)co 7381  ℂcc 11057  ℝcr 11058  0cc0 11059  ici 11061   · cmul 11064  +∞cpnf 11199  ℝ^*cxr 11201   ≤ cle 11203   / cdiv 11830  3c3 12259  ℤcz 12554  [,]cicc 13338  ...cfz 13498  ↑cexp 14060  ℜcre 15096  Σcsu 15685  vol*covol 25493  MblFncmbf 25645  ∫₂citg2 25647  𝐿¹cibl 25648  ∫citg 25649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-symdif 4196  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-disj 5058  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-of 7645  df-ofr 7646  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-er 8662  df-map 8794  df-pm 8795  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-fi 9343  df-sup 9374  df-inf 9375  df-oi 9444  df-dju 9845  df-card 9883  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-n0 12468  df-z 12555  df-uz 12826  df-q 12936  df-rp 12980  df-xneg 13100  df-xadd 13101  df-xmul 13102  df-ioo 13339  df-ico 13341  df-icc 13342  df-fz 13499  df-fzo 13646  df-fl 13788  df-seq 14001  df-exp 14061  df-hash 14330  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-clim 15487  df-sum 15686  df-rest 17423  df-topgen 17444  df-psmet 21385  df-xmet 21386  df-met 21387  df-bl 21388  df-mopn 21389  df-top 22923  df-topon 22940  df-bases 22975  df-cmp 23416  df-ovol 25495  df-vol 25496  df-mbf 25650  df-itg1 25651  df-itg2 25652  df-ibl 25653  df-itg 25654

This theorem is referenced by:  itgss3  25846