Theorem itgeqa 23580

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 23410	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ MblFn ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ MblFn))
7		ifan 4134	. . . . . . . . . 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 751	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ ℂ)
9		elfzelz 12342	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (0...3) → 𝑘 ∈ ℤ)
10	9	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝑘 ∈ ℤ)
11		ax-icn 9995	. . . . . . . . . . . . . . . . . 18 ⊢ i ∈ ℂ
12		ine0 10465	. . . . . . . . . . . . . . . . . 18 ⊢ i ≠ 0
13		expclz 12885	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ∈ ℂ)
14	11, 12, 13	mp3an12 1414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ∈ ℂ)
15	10, 14	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ∈ ℂ)
16		expne0i 12892	. . . . . . . . . . . . . . . . . 18 ⊢ ((i ∈ ℂ ∧ i ≠ 0 ∧ 𝑘 ∈ ℤ) → (i↑𝑘) ≠ 0)
17	11, 12, 16	mp3an12 1414	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℤ → (i↑𝑘) ≠ 0)
18	10, 17	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (i↑𝑘) ≠ 0)
19	8, 15, 18	divcld 10801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) ∈ ℂ)
20	19	recld 13934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ)
21		0re 10040	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
22		ifcl 4130	. . . . . . . . . . . . . 14 ⊢ (((ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
23	20, 21, 22	sylancl 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ)
24	23	rexrd 10089	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ ℝ*)
25		max1 12016	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (ℜ‘(𝐶 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
26	21, 20, 25	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0))
27		elxrge0 12281	. . . . . . . . . . . 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)))
28	24, 26, 27	sylanbrc 698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
29		0e0iccpnf 12283	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]+∞)
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ ¬ 𝑥 ∈ 𝐵) → 0 ∈ (0[,]+∞))
31	28, 30	ifclda 4120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐶 / (i↑𝑘))), (ℜ‘(𝐶 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
32	7, 31	syl5eqel 2705	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
33	32	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ (0[,]+∞))
34		eqid 2622	. . . . . . . 8 ⊢ (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))
35	33, 34	fmptd 6385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
36		ifan 4134	. . . . . . . . . 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)
37	5	adantlr 751	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 𝐷 ∈ ℂ)
38	37, 15, 18	divcld 10801	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (𝐷 / (i↑𝑘)) ∈ ℂ)
39	38	recld 13934	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ)
40		ifcl 4130	. . . . . . . . . . . . . 14 ⊢ (((ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
41	39, 21, 40	sylancl 694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ)
42	41	rexrd 10089	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ ℝ*)
43		max1 12016	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ (ℜ‘(𝐷 / (i↑𝑘))) ∈ ℝ) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
44	21, 39, 43	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → 0 ≤ if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0))
45		elxrge0 12281	. . . . . . . . . . . 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)))
46	42, 44, 45	sylanbrc 698	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ 𝐵) → if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
47	46, 30	ifclda 4120	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if(𝑥 ∈ 𝐵, if(0 ≤ (ℜ‘(𝐷 / (i↑𝑘))), (ℜ‘(𝐷 / (i↑𝑘))), 0), 0) ∈ (0[,]+∞))
48	36, 47	syl5eqel 2705	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
49	48	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0...3)) ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ (0[,]+∞))
50		eqid 2622	. . . . . . . 8 ⊢ (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))
51	49, 50	fmptd 6385	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0)):ℝ⟶(0[,]+∞))
52	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → 𝐴 ⊆ ℝ)
53	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...3)) → (vol*‘𝐴) = 0)
54		simpll 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝜑)
55		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
56		eldifn 3733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
57	56	ad2antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ 𝐴)
58	55, 57	eldifd 3585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∖ 𝐴))
59	54, 58, 3	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)
60	59	oveq1d 6665	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → (𝐶 / (i↑𝑘)) = (𝐷 / (i↑𝑘)))
61	60	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
62	61	ibllem 23531	. . . . . . . . . . . 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))
63		eldifi 3732	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → 𝑥 ∈ ℝ)
64	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 𝑥 ∈ ℝ)
65		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) ∈ V
66		c0ex 10034	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
67	65, 66	ifex 4156	. . . . . . . . . . . . 13 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0) ∈ V
68	34	fvmpt2 6291	. . . . . . . . . . . . 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))
69	64, 67, 68	sylancl 694	. . . . . . . . . . . 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))
70		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (ℜ‘(𝐷 / (i↑𝑘))) ∈ V
71	70, 66	ifex 4156	. . . . . . . . . . . . 13 ⊢ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0) ∈ V
72	50	fvmpt2 6291	. . . . . . . . . . . . 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))
73	64, 71, 72	sylancl 694	. . . . . . . . . . . 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))
74	62, 69, 73	3eqtr4d 2666	. . . . . . . . . . 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))
75	74	ralrimiva 2966	. . . . . . . . . 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))
76		nfv 1843	. . . . . . . . . . 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)
77		nffvmpt1 6199	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))‘𝑦)
78		nffvmpt1 6199	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))‘𝑦)
79	77, 78	nfeq 2776	. . . . . . . . . . 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)
80		fveq2 6191	. . . . . . . . . . . 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))
81		fveq2 6191	. . . . . . . . . . . 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))
82	80, 81	eqeq12d 2637	. . . . . . . . . . 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)))
83	76, 79, 82	cbvral 3167	. . . . . . . . . 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))
84	75, 83	sylib 208	. . . . . . . . 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))
85	84	r19.21bi 2932	. . . . . . . 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))
86	85	adantlr 751	. . . . . . 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))
87	35, 51, 52, 53, 86	itg2eqa 23512	. . . . . 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))))
88	87	eleq1d 2686	. . . . 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))
89	88	ralbidva 2985	. . . 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))
90	6, 89	anbi12d 747	. . 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)))
91		eqidd 2623	. . . 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)))
92		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘))))
93	91, 92, 4	isibl2 23533	. . 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)))
94		eqidd 2623	. . . 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)))
95		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘))))
96	94, 95, 5	isibl2 23533	. . 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)))
97	90, 93, 96	3bitr4d 300	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1))
98	87	oveq2d 6666	. . . 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)))))
99	98	sumeq2dv 14433	. . 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)))))
100		eqid 2622	. . . 4 ⊢ (ℜ‘(𝐶 / (i↑𝑘))) = (ℜ‘(𝐶 / (i↑𝑘)))
101	100	dfitg 23536	. . 3 ⊢ ∫𝐵𝐶 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐶 / (i↑𝑘)))), (ℜ‘(𝐶 / (i↑𝑘))), 0))))
102		eqid 2622	. . . 4 ⊢ (ℜ‘(𝐷 / (i↑𝑘))) = (ℜ‘(𝐷 / (i↑𝑘)))
103	102	dfitg 23536	. . 3 ⊢ ∫𝐵𝐷 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐵 ∧ 0 ≤ (ℜ‘(𝐷 / (i↑𝑘)))), (ℜ‘(𝐷 / (i↑𝑘))), 0))))
104	99, 101, 103	3eqtr4g 2681	. 2 ⊢ (𝜑 → ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
105	97, 104	jca 554	1 ⊢ (𝜑 → (((𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1 ↔ (𝑥 ∈ 𝐵 ↦ 𝐷) ∈ 𝐿1) ∧ ∫𝐵𝐶 d𝑥 = ∫𝐵𝐷 d𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  ici 9938   · cmul 9941  +∞cpnf 10071  ℝ^*cxr 10073   ≤ cle 10075   / cdiv 10684  3c3 11071  ℤcz 11377  [,]cicc 12178  ...cfz 12326  ↑cexp 12860  ℜcre 13837  Σcsu 14416  vol*covol 23231  MblFncmbf 23383  ∫₂citg2 23385  𝐿¹cibl 23386  ∫citg 23387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392

This theorem is referenced by:  itgss3  23581