itgmulc2nclem2 - Mathbox for Brendan Leahy

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Theorem itgmulc2nclem2 37213

Description: Lemma for itgmulc2nc 37214; cf. itgmulc2lem2 25775. (Contributed by Brendan Leahy, 19-Nov-2017.)

**Hypotheses**
Ref	Expression
itgmulc2nc.1	⊢ (𝜑 → 𝐶 ∈ ℂ)
itgmulc2nc.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgmulc2nc.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
itgmulc2nc.m	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
itgmulc2nc.4	⊢ (𝜑 → 𝐶 ∈ ℝ)
itgmulc2nc.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
itgmulc2nclem2	⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐶 𝜑,𝑥 𝑥,𝑉 Allowed substitution hint: 𝐵(𝑥)

**Proof of Theorem itgmulc2nclem2**
Step	Hyp	Ref	Expression
1		itgmulc2nc.4	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
2		max0sub 13202	. . . . . . 7 ⊢ (𝐶 ∈ ℝ → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) = 𝐶)
4	3	oveq1d 7428	. . . . 5 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
5	4	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = (𝐶 · 𝐵))
6		0re 11241	. . . . . . . 8 ⊢ 0 ∈ ℝ
7		ifcl 4570	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
8	1, 6, 7	sylancl 584	. . . . . . 7 ⊢ (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
9	8	recnd 11267	. . . . . 6 ⊢ (𝜑 → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
10	9	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℂ)
11	1	renegcld 11666	. . . . . . . 8 ⊢ (𝜑 → -𝐶 ∈ ℝ)
12		ifcl 4570	. . . . . . . 8 ⊢ ((-𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
13	11, 6, 12	sylancl 584	. . . . . . 7 ⊢ (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
14	13	recnd 11267	. . . . . 6 ⊢ (𝜑 → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
15	14	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℂ)
16		itgmulc2nc.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
17	16	recnd 11267	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
18	10, 15, 17	subdird 11696	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
19	5, 18	eqtr3d 2767	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
20	19	itgeq2dv 25724	. 2 ⊢ (𝜑 → ∫𝐴(𝐶 · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥)
21		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) ∈ V)
22		itgmulc2nc.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
23		itgmulc2nc.m	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
24		ovexd 7448	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 · 𝐵) ∈ V)
25	23, 24	mbfdm2 25579	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom vol)
26	8	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
27		fconstmpt 5735	. . . . . . 7 ⊢ (𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0))
28	27	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)))
29		eqidd 2726	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵))
30	25, 26, 16, 28, 29	offval2 7699	. . . . 5 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)))
31		iblmbf 25710	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
32	22, 31	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
33	17	fmpttd 7118	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
34	32, 8, 33	mbfmulc2re 25590	. . . . 5 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn)
35	30, 34	eqeltrrd 2826	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ MblFn)
36	9, 16, 22, 35	iblmulc2nc 37211	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) ∈ 𝐿¹)
37		ovexd 7448	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) ∈ V)
38	13	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐶, -𝐶, 0) ∈ ℝ)
39		fconstmpt 5735	. . . . . . 7 ⊢ (𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0))
40	39	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐶, -𝐶, 0)))
41	25, 38, 16, 40, 29	offval2 7699	. . . . 5 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)))
42	32, 13, 33	mbfmulc2re 25590	. . . . 5 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ MblFn)
43	41, 42	eqeltrrd 2826	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ MblFn)
44	14, 16, 22, 43	iblmulc2nc 37211	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) ∈ 𝐿¹)
45	19	mpteq2dva 5244	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))))
46	45, 23	eqeltrrd 2826	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))) ∈ MblFn)
47	21, 36, 37, 44, 46	itgsubnc 37208	. 2 ⊢ (𝜑 → ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) − (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) d𝑥 = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥))
48		ovexd 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
49		ifcl 4570	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
50	16, 6, 49	sylancl 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
51	16	iblre 25736	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿¹)))
52	22, 51	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿¹))
53	52	simpld 493	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿¹)
54		eqidd 2726	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)))
55	25, 26, 50, 28, 54	offval2 7699	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))))
56	16, 32	mbfpos 25593	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
57	50	recnd 11267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ)
58	57	fmpttd 7118	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)):𝐴⟶ℂ)
59	56, 8, 58	mbfmulc2re 25590	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
60	55, 59	eqeltrrd 2826	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
61	9, 50, 53, 60	iblmulc2nc 37211	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿¹)
62		ovexd 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
63	16	renegcld 11666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝐵 ∈ ℝ)
64		ifcl 4570	. . . . . . . 8 ⊢ ((-𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
65	63, 6, 64	sylancl 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℝ)
66	52	simprd 494	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿¹)
67		eqidd 2726	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)))
68	25, 26, 65, 28, 67	offval2 7699	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
69	16, 32	mbfneg 25592	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
70	63, 69	mbfpos 25593	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)
71	65	recnd 11267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ -𝐵, -𝐵, 0) ∈ ℂ)
72	71	fmpttd 7118	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)):𝐴⟶ℂ)
73	70, 8, 72	mbfmulc2re 25590	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
74	68, 73	eqeltrrd 2826	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
75	9, 65, 66, 74	iblmulc2nc 37211	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿¹)
76		max0sub 13202	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
77	16, 76	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0)) = 𝐵)
78	77	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵))
79	10, 57, 71	subdid 11695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
80	78, 79	eqtr3d 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) = ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
81	80	mpteq2dva 5244	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐶, 𝐶, 0) · 𝐵)) = (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
82	30, 81	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ 𝐶, 𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
83	82, 34	eqeltrrd 2826	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))) ∈ MblFn)
84	48, 61, 62, 75, 83	itgsubnc 37208	. . . . 5 ⊢ (𝜑 → ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥 = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
85	80	itgeq2dv 25724	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥)
86	16, 22	itgreval 25739	. . . . . . 7 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
87	86	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) = (if(0 ≤ 𝐶, 𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
88	50, 53	itgcl 25726	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 ∈ ℂ)
89	65, 66	itgcl 25726	. . . . . . 7 ⊢ (𝜑 → ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥 ∈ ℂ)
90	9, 88, 89	subdid 11695	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
91		max1 13191	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
92	6, 1, 91	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
93		max1 13191	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
94	6, 16, 93	sylancr 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
95	9, 50, 53, 60, 8, 50, 92, 94	itgmulc2nclem1 37212	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥)
96		max1 13191	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ -𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
97	6, 63, 96	sylancr 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ -𝐵, -𝐵, 0))
98	9, 65, 66, 74, 8, 65, 92, 97	itgmulc2nclem1 37212	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥)
99	95, 98	oveq12d 7431	. . . . . 6 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
100	87, 90, 99	3eqtrd 2769	. . . . 5 ⊢ (𝜑 → (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) = (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
101	84, 85, 100	3eqtr4d 2775	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥))
102		ovexd 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) ∈ V)
103	25, 38, 50, 40, 54	offval2 7699	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))))
104	56, 13, 58	mbfmulc2re 25590	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
105	103, 104	eqeltrrd 2826	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ MblFn)
106	14, 50, 53, 105	iblmulc2nc 37211	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0))) ∈ 𝐿¹)
107		ovexd 7448	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) ∈ V)
108	25, 38, 65, 40, 67	offval2 7699	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) = (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
109	70, 13, 72	mbfmulc2re 25590	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
110	108, 109	eqeltrrd 2826	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ MblFn)
111	14, 65, 66, 110	iblmulc2nc 37211	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) ∈ 𝐿¹)
112	77	oveq2d 7429	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵))
113	15, 57, 71	subdid 11695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · (if(0 ≤ 𝐵, 𝐵, 0) − if(0 ≤ -𝐵, -𝐵, 0))) = ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
114	112, 113	eqtr3d 2767	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) = ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))))
115	114	mpteq2dva 5244	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ -𝐶, -𝐶, 0) · 𝐵)) = (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
116	41, 115	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × {if(0 ≤ -𝐶, -𝐶, 0)}) ∘_f · (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))))
117	116, 42	eqeltrrd 2826	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)))) ∈ MblFn)
118	102, 106, 107, 111, 117	itgsubnc 37208	. . . . 5 ⊢ (𝜑 → ∫𝐴((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥 = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
119	114	itgeq2dv 25724	. . . . 5 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = ∫𝐴((if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) − (if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0))) d𝑥)
120	86	oveq2d 7429	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥) = (if(0 ≤ -𝐶, -𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
121	14, 88, 89	subdid 11695	. . . . . 6 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = ((if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)))
122		max1 13191	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ -𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
123	6, 11, 122	sylancr 585	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ if(0 ≤ -𝐶, -𝐶, 0))
124	14, 50, 53, 105, 13, 50, 123, 94	itgmulc2nclem1 37212	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥)
125	14, 65, 66, 110, 13, 65, 123, 97	itgmulc2nclem1 37212	. . . . . . 7 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥) = ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥)
126	124, 125	oveq12d 7431	. . . . . 6 ⊢ (𝜑 → ((if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥)) = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
127	120, 121, 126	3eqtrd 2769	. . . . 5 ⊢ (𝜑 → (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥) = (∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ 𝐵, 𝐵, 0)) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · if(0 ≤ -𝐵, -𝐵, 0)) d𝑥))
128	118, 119, 127	3eqtr4d 2775	. . . 4 ⊢ (𝜑 → ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥 = (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥))
129	101, 128	oveq12d 7431	. . 3 ⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥)))
130	16, 22	itgcl 25726	. . . 4 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
131	9, 14, 130	subdird 11696	. . 3 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = ((if(0 ≤ 𝐶, 𝐶, 0) · ∫𝐴𝐵 d𝑥) − (if(0 ≤ -𝐶, -𝐶, 0) · ∫𝐴𝐵 d𝑥)))
132	3	oveq1d 7428	. . 3 ⊢ (𝜑 → ((if(0 ≤ 𝐶, 𝐶, 0) − if(0 ≤ -𝐶, -𝐶, 0)) · ∫𝐴𝐵 d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
133	129, 131, 132	3eqtr2d 2771	. 2 ⊢ (𝜑 → (∫𝐴(if(0 ≤ 𝐶, 𝐶, 0) · 𝐵) d𝑥 − ∫𝐴(if(0 ≤ -𝐶, -𝐶, 0) · 𝐵) d𝑥) = (𝐶 · ∫𝐴𝐵 d𝑥))
134	20, 47, 133	3eqtrrd 2770	1 ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ifcif 4525 {csn 4625 class class class wbr 5144 ↦ cmpt 5227 × cxp 5671 dom cdm 5673 (class class class)co 7413 ∘_f cof 7677 ℂcc 11131 ℝcr 11132 0cc0 11133 · cmul 11138 ≤ cle 11274 − cmin 11469 -cneg 11470 volcvol 25405 MblFncmbf 25556 𝐿¹cibl 25559 ∫citg 25560

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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-sum 15660 df-rest 17398 df-topgen 17419 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-top 22809 df-topon 22826 df-bases 22862 df-cmp 23304 df-ovol 25406 df-vol 25407 df-mbf 25561 df-itg1 25562 df-itg2 25563 df-ibl 25564 df-itg 25565 df-0p 25612

This theorem is referenced by: itgmulc2nc 37214

W3C validator