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Theorem itgadd 25676

Description: Add two integrals over the same domain. (Contributed by Mario Carneiro, 17-Aug-2014.)

**Hypotheses**
Ref	Expression
itgadd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
itgadd.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
itgadd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
itgadd.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)

**Assertion**
Ref	Expression
itgadd	⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))

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

**Proof of Theorem itgadd**
Step	Hyp	Ref	Expression
1		itgadd.2	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹)
2		iblmbf 25619	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4		itgadd.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
5	3, 4	mbfmptcl 25487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
6		itgadd.4	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹)
7		iblmbf 25619	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
8	6, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
9		itgadd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)
10	8, 9	mbfmptcl 25487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)
11	5, 10	readdd 15158	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘(𝐵 + 𝐶)) = ((ℜ‘𝐵) + (ℜ‘𝐶)))
12	11	itgeq2dv 25633	. . . . 5 ⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥)
13	5	recld 15138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐵) ∈ ℝ)
14	5	iblcn 25650	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)))
15	1, 14	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹))
16	15	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿¹)
17	10	recld 15138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℜ‘𝐶) ∈ ℝ)
18	10	iblcn 25650	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈ 𝐿¹)))
19	6, 18	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿¹ ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈ 𝐿¹))
20	19	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ 𝐿¹)
21	13, 16, 17, 20, 13, 17	itgaddlem2 25675	. . . . 5 ⊢ (𝜑 → ∫𝐴((ℜ‘𝐵) + (ℜ‘𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥))
22	12, 21	eqtrd 2764	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥))
23	5, 10	imaddd 15159	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘(𝐵 + 𝐶)) = ((ℑ‘𝐵) + (ℑ‘𝐶)))
24	23	itgeq2dv 25633	. . . . . . 7 ⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥)
25	5	imcld 15139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐵) ∈ ℝ)
26	15	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿¹)
27	10	imcld 15139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (ℑ‘𝐶) ∈ ℝ)
28	19	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈ 𝐿¹)
29	25, 26, 27, 28, 25, 27	itgaddlem2 25675	. . . . . . 7 ⊢ (𝜑 → ∫𝐴((ℑ‘𝐵) + (ℑ‘𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥))
30	24, 29	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥 = (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥))
31	30	oveq2d 7417	. . . . 5 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)))
32		ax-icn 11165	. . . . . . 7 ⊢ i ∈ ℂ
33	32	a1i 11	. . . . . 6 ⊢ (𝜑 → i ∈ ℂ)
34	25, 26	itgcl 25635	. . . . . 6 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ)
35	27, 28	itgcl 25635	. . . . . 6 ⊢ (𝜑 → ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ)
36	33, 34, 35	adddid 11235	. . . . 5 ⊢ (𝜑 → (i · (∫𝐴(ℑ‘𝐵) d𝑥 + ∫𝐴(ℑ‘𝐶) d𝑥)) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))
37	31, 36	eqtrd 2764	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥) = ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))
38	22, 37	oveq12d 7419	. . 3 ⊢ (𝜑 → (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥)) = ((∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥) + ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))))
39	13, 16	itgcl 25635	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐵) d𝑥 ∈ ℂ)
40	17, 20	itgcl 25635	. . . 4 ⊢ (𝜑 → ∫𝐴(ℜ‘𝐶) d𝑥 ∈ ℂ)
41		mulcl 11190	. . . . 5 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐵) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
42	32, 34, 41	sylancr 586	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐵) d𝑥) ∈ ℂ)
43		mulcl 11190	. . . . 5 ⊢ ((i ∈ ℂ ∧ ∫𝐴(ℑ‘𝐶) d𝑥 ∈ ℂ) → (i · ∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ)
44	32, 35, 43	sylancr 586	. . . 4 ⊢ (𝜑 → (i · ∫𝐴(ℑ‘𝐶) d𝑥) ∈ ℂ)
45	39, 40, 42, 44	add4d 11439	. . 3 ⊢ (𝜑 → ((∫𝐴(ℜ‘𝐵) d𝑥 + ∫𝐴(ℜ‘𝐶) d𝑥) + ((i · ∫𝐴(ℑ‘𝐵) d𝑥) + (i · ∫𝐴(ℑ‘𝐶) d𝑥))) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))))
46	38, 45	eqtrd 2764	. 2 ⊢ (𝜑 → (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥)) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))))
47		ovexd 7436	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ V)
48	4, 1, 9, 6	ibladd 25672	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿¹)
49	47, 48	itgcnval 25651	. 2 ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴(ℜ‘(𝐵 + 𝐶)) d𝑥 + (i · ∫𝐴(ℑ‘(𝐵 + 𝐶)) d𝑥)))
50	4, 1	itgcnval 25651	. . 3 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)))
51	9, 6	itgcnval 25651	. . 3 ⊢ (𝜑 → ∫𝐴𝐶 d𝑥 = (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥)))
52	50, 51	oveq12d 7419	. 2 ⊢ (𝜑 → (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥) = ((∫𝐴(ℜ‘𝐵) d𝑥 + (i · ∫𝐴(ℑ‘𝐵) d𝑥)) + (∫𝐴(ℜ‘𝐶) d𝑥 + (i · ∫𝐴(ℑ‘𝐶) d𝑥))))
53	46, 49, 52	3eqtr4d 2774	1 ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 ℂcc 11104 ℝcr 11105 ici 11108 + caddc 11109 · cmul 11111 ℜcre 15041 ℑcim 15042 MblFncmbf 25465 𝐿¹cibl 25468 ∫citg 25469

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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-disj 5104 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-rest 17367 df-topgen 17388 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-top 22718 df-topon 22735 df-bases 22771 df-cmp 23213 df-ovol 25315 df-vol 25316 df-mbf 25470 df-itg1 25471 df-itg2 25472 df-ibl 25473 df-itg 25474 df-0p 25521

This theorem is referenced by: itgsub 25677 itgfsum 25678 itgmulc2 25685 ftc1lem4 25896 itgparts 25904 areaquad 42454 fourierdlem83 45390 fourierdlem95 45402

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