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Theorem itgconst 25768

Description: Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Assertion**
Ref	Expression
itgconst	⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴𝐵 d𝑥 = (𝐵 · (vol‘𝐴)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem itgconst Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 481	. . . . . . 7 ⊢ ((𝑦 = (ℜ‘𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 = (ℜ‘𝐵))
2	1	itgeq2dv 25731	. . . . . 6 ⊢ (𝑦 = (ℜ‘𝐵) → ∫𝐴𝑦 d𝑥 = ∫𝐴(ℜ‘𝐵) d𝑥)
3		oveq1 7433	. . . . . 6 ⊢ (𝑦 = (ℜ‘𝐵) → (𝑦 · (vol‘𝐴)) = ((ℜ‘𝐵) · (vol‘𝐴)))
4	2, 3	eqeq12d 2744	. . . . 5 ⊢ (𝑦 = (ℜ‘𝐵) → (∫𝐴𝑦 d𝑥 = (𝑦 · (vol‘𝐴)) ↔ ∫𝐴(ℜ‘𝐵) d𝑥 = ((ℜ‘𝐵) · (vol‘𝐴))))
5		simplr 767	. . . . . . . 8 ⊢ ((((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ ℝ)
6		fconstmpt 5744	. . . . . . . . 9 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦)
7		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → 𝐴 ∈ dom vol)
8		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (vol‘𝐴) ∈ ℝ)
9	8	adantr 479	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (vol‘𝐴) ∈ ℝ)
10		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
11	10	recnd 11280	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
12		iblconst 25767	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℂ) → (𝐴 × {𝑦}) ∈ 𝐿¹)
13	7, 9, 11, 12	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (𝐴 × {𝑦}) ∈ 𝐿¹)
14	6, 13	eqeltrrid 2834	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ 𝐿¹)
15	5, 14	itgrevallem1 25744	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → ∫𝐴𝑦 d𝑥 = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0)))))
16		ifan 4585	. . . . . . . . . . . 12 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝑦, 𝑦, 0), 0)
17	16	mpteq2i 5257	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ 𝑦, 𝑦, 0), 0))
18	17	fveq2i 6905	. . . . . . . . . 10 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ 𝑦, 𝑦, 0), 0)))
19		0re 11254	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
20		ifcl 4577	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
21	10, 19, 20	sylancl 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ)
22		max1 13204	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
23	19, 10, 22	sylancr 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ 𝑦, 𝑦, 0))
24		elrege0 13471	. . . . . . . . . . . 12 ⊢ (if(0 ≤ 𝑦, 𝑦, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝑦, 𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝑦, 𝑦, 0)))
25	21, 23, 24	sylanbrc 581	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ (0[,)+∞))
26		itg2const 25690	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ if(0 ≤ 𝑦, 𝑦, 0) ∈ (0[,)+∞)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ 𝑦, 𝑦, 0), 0))) = (if(0 ≤ 𝑦, 𝑦, 0) · (vol‘𝐴)))
27	7, 9, 25, 26	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ 𝑦, 𝑦, 0), 0))) = (if(0 ≤ 𝑦, 𝑦, 0) · (vol‘𝐴)))
28	18, 27	eqtrid 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) = (if(0 ≤ 𝑦, 𝑦, 0) · (vol‘𝐴)))
29		ifan 4585	. . . . . . . . . . . 12 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ -𝑦, -𝑦, 0), 0)
30	29	mpteq2i 5257	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ -𝑦, -𝑦, 0), 0))
31	30	fveq2i 6905	. . . . . . . . . 10 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ -𝑦, -𝑦, 0), 0)))
32		renegcl 11561	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
33	32	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → -𝑦 ∈ ℝ)
34		ifcl 4577	. . . . . . . . . . . . 13 ⊢ ((-𝑦 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ -𝑦, -𝑦, 0) ∈ ℝ)
35	33, 19, 34	sylancl 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ -𝑦, -𝑦, 0) ∈ ℝ)
36		max1 13204	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ -𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -𝑦, -𝑦, 0))
37	19, 33, 36	sylancr 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → 0 ≤ if(0 ≤ -𝑦, -𝑦, 0))
38		elrege0 13471	. . . . . . . . . . . 12 ⊢ (if(0 ≤ -𝑦, -𝑦, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ -𝑦, -𝑦, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ -𝑦, -𝑦, 0)))
39	35, 37, 38	sylanbrc 581	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ -𝑦, -𝑦, 0) ∈ (0[,)+∞))
40		itg2const 25690	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ if(0 ≤ -𝑦, -𝑦, 0) ∈ (0[,)+∞)) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ -𝑦, -𝑦, 0), 0))) = (if(0 ≤ -𝑦, -𝑦, 0) · (vol‘𝐴)))
41	7, 9, 39, 40	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, if(0 ≤ -𝑦, -𝑦, 0), 0))) = (if(0 ≤ -𝑦, -𝑦, 0) · (vol‘𝐴)))
42	31, 41	eqtrid 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0))) = (if(0 ≤ -𝑦, -𝑦, 0) · (vol‘𝐴)))
43	28, 42	oveq12d 7444	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0)))) = ((if(0 ≤ 𝑦, 𝑦, 0) · (vol‘𝐴)) − (if(0 ≤ -𝑦, -𝑦, 0) · (vol‘𝐴))))
44	21	recnd 11280	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ 𝑦, 𝑦, 0) ∈ ℂ)
45	35	recnd 11280	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → if(0 ≤ -𝑦, -𝑦, 0) ∈ ℂ)
46	8	recnd 11280	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (vol‘𝐴) ∈ ℂ)
47	46	adantr 479	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (vol‘𝐴) ∈ ℂ)
48	44, 45, 47	subdird 11709	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → ((if(0 ≤ 𝑦, 𝑦, 0) − if(0 ≤ -𝑦, -𝑦, 0)) · (vol‘𝐴)) = ((if(0 ≤ 𝑦, 𝑦, 0) · (vol‘𝐴)) − (if(0 ≤ -𝑦, -𝑦, 0) · (vol‘𝐴))))
49		max0sub 13215	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → (if(0 ≤ 𝑦, 𝑦, 0) − if(0 ≤ -𝑦, -𝑦, 0)) = 𝑦)
50	49	adantl 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (if(0 ≤ 𝑦, 𝑦, 0) − if(0 ≤ -𝑦, -𝑦, 0)) = 𝑦)
51	50	oveq1d 7441	. . . . . . . 8 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → ((if(0 ≤ 𝑦, 𝑦, 0) − if(0 ≤ -𝑦, -𝑦, 0)) · (vol‘𝐴)) = (𝑦 · (vol‘𝐴)))
52	43, 48, 51	3eqtr2rd 2775	. . . . . . 7 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → (𝑦 · (vol‘𝐴)) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))) − (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝑦), -𝑦, 0)))))
53	15, 52	eqtr4d 2771	. . . . . 6 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑦 ∈ ℝ) → ∫𝐴𝑦 d𝑥 = (𝑦 · (vol‘𝐴)))
54	53	ralrimiva 3143	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∀𝑦 ∈ ℝ ∫𝐴𝑦 d𝑥 = (𝑦 · (vol‘𝐴)))
55		recl 15097	. . . . . 6 ⊢ (𝐵 ∈ ℂ → (ℜ‘𝐵) ∈ ℝ)
56	55	3ad2ant3 1132	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℝ)
57	4, 54, 56	rspcdva 3612	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴(ℜ‘𝐵) d𝑥 = ((ℜ‘𝐵) · (vol‘𝐴)))
58		simpl 481	. . . . . . . . 9 ⊢ ((𝑦 = (ℑ‘𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 = (ℑ‘𝐵))
59	58	itgeq2dv 25731	. . . . . . . 8 ⊢ (𝑦 = (ℑ‘𝐵) → ∫𝐴𝑦 d𝑥 = ∫𝐴(ℑ‘𝐵) d𝑥)
60		oveq1 7433	. . . . . . . 8 ⊢ (𝑦 = (ℑ‘𝐵) → (𝑦 · (vol‘𝐴)) = ((ℑ‘𝐵) · (vol‘𝐴)))
61	59, 60	eqeq12d 2744	. . . . . . 7 ⊢ (𝑦 = (ℑ‘𝐵) → (∫𝐴𝑦 d𝑥 = (𝑦 · (vol‘𝐴)) ↔ ∫𝐴(ℑ‘𝐵) d𝑥 = ((ℑ‘𝐵) · (vol‘𝐴))))
62		imcl 15098	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) ∈ ℝ)
63	62	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘𝐵) ∈ ℝ)
64	61, 54, 63	rspcdva 3612	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴(ℑ‘𝐵) d𝑥 = ((ℑ‘𝐵) · (vol‘𝐴)))
65	64	oveq2d 7442	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) = (i · ((ℑ‘𝐵) · (vol‘𝐴))))
66		ax-icn 11205	. . . . . . 7 ⊢ i ∈ ℂ
67	66	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → i ∈ ℂ)
68	63	recnd 11280	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℑ‘𝐵) ∈ ℂ)
69	67, 68, 46	mulassd 11275	. . . . 5 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ((i · (ℑ‘𝐵)) · (vol‘𝐴)) = (i · ((ℑ‘𝐵) · (vol‘𝐴))))
70	65, 69	eqtr4d 2771	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) = ((i · (ℑ‘𝐵)) · (vol‘𝐴)))
71	57, 70	oveq12d 7444	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (((ℜ‘𝐵) · (vol‘𝐴)) + ((i · (ℑ‘𝐵)) · (vol‘𝐴))))
72	56	recnd 11280	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (ℜ‘𝐵) ∈ ℂ)
73		mulcl 11230	. . . . 5 ⊢ ((i ∈ ℂ ∧ (ℑ‘𝐵) ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
74	66, 68, 73	sylancr 585	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (i · (ℑ‘𝐵)) ∈ ℂ)
75	72, 74, 46	adddird 11277	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (((ℜ‘𝐵) + (i · (ℑ‘𝐵))) · (vol‘𝐴)) = (((ℜ‘𝐵) · (vol‘𝐴)) + ((i · (ℑ‘𝐵)) · (vol‘𝐴))))
76	71, 75	eqtr4d 2771	. 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) = (((ℜ‘𝐵) + (i · (ℑ‘𝐵))) · (vol‘𝐴)))
77		simpl3 1190	. . 3 ⊢ (((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
78		fconstmpt 5744	. . . 4 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
79		iblconst 25767	. . . 4 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐴 × {𝐵}) ∈ 𝐿¹)
80	78, 79	eqeltrrid 2834	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
81	77, 80	itgcnval 25749	. 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
82		replim 15103	. . . 4 ⊢ (𝐵 ∈ ℂ → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
83	82	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → 𝐵 = ((ℜ‘𝐵) + (i · (ℑ‘𝐵))))
84	83	oveq1d 7441	. 2 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → (𝐵 · (vol‘𝐴)) = (((ℜ‘𝐵) + (i · (ℑ‘𝐵))) · (vol‘𝐴)))
85	76, 81, 84	3eqtr4d 2778	1 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℂ) → ∫𝐴𝐵 d𝑥 = (𝐵 · (vol‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ifcif 4532 {csn 4632 class class class wbr 5152 ↦ cmpt 5235 × cxp 5680 dom cdm 5682 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 ici 11148 + caddc 11149 · cmul 11151 +∞cpnf 11283 ≤ cle 11287 − cmin 11482 -cneg 11483 [,)cico 13366 ℜcre 15084 ℑcim 15085 volcvol 25412 ∫₂citg2 25565 𝐿¹cibl 25566 ∫citg 25567

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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 df-itg2 25570 df-ibl 25571 df-itg 25572 df-0p 25619

This theorem is referenced by: ftc1lem4 25994 itgulm 26364 itgexpif 34271 ftc1cnnclem 37197 arearect 42674 areaquad 42675 wallispilem2 45483 fourierdlem87 45610 sqwvfoura 45645 etransclem23 45674

W3C validator