3factsumint1 - Mathbox for metakunt

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Theorem 3factsumint1 42113

Description: Move constants out of integrals or sums and/or commute sum and integral. (Contributed by metakunt, 26-Apr-2024.)

**Hypotheses**
Ref	Expression
3factsumint1.1	⊢ 𝐴 = (𝐿[,]𝑈)
3factsumint1.2	⊢ (𝜑 → 𝐵 ∈ Fin)
3factsumint1.3	⊢ (𝜑 → 𝐿 ∈ ℝ)
3factsumint1.4	⊢ (𝜑 → 𝑈 ∈ ℝ)
3factsumint1.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
3factsumint1.6	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))
3factsumint1.7	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
3factsumint1.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
3factsumint1.9	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))

**Assertion**
Ref	Expression
3factsumint1	⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)

Distinct variable groups: 𝐴,𝑘,𝑥 𝐵,𝑘,𝑥 𝑥,𝐺 𝜑,𝑘,𝑥 Allowed substitution hints: 𝑈(𝑥,𝑘) 𝐹(𝑥,𝑘) 𝐺(𝑘) 𝐻(𝑥,𝑘) 𝐿(𝑥,𝑘)

Proof of Theorem 3factsumint1 Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3factsumint1.1	. . . 4 ⊢ 𝐴 = (𝐿[,]𝑈)
2		3factsumint1.3	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ℝ)
3		3factsumint1.4	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
4		iccmbl 25494	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐿[,]𝑈) ∈ dom vol)
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐿[,]𝑈) ∈ dom vol)
6	1, 5	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
7		3factsumint1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		3factsumint1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
9	8	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐹 ∈ ℂ)
10		3factsumint1.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
11	10	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐺 ∈ ℂ)
12		3factsumint1.8	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
13	11, 12	mulcld 11132	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐺 · 𝐻) ∈ ℂ)
14	9, 13	mulcld 11132	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐹 · (𝐺 · 𝐻)) ∈ ℂ)
15		ovex 7379	. . . . . . 7 ⊢ (𝐿[,]𝑈) ∈ V
16	1, 15	eqeltri 2827	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ∈ V)
18	9	anass1rs 655	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
19	13	anass1rs 655	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 · 𝐻) ∈ ℂ)
20		eqidd 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹))
21		eqidd 2732	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) = (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)))
22	17, 18, 19, 20, 21	offval2 7630	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) = (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))))
23		3factsumint1.6	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))
24		cnmbf 25587	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
25	6, 23, 24	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
27	12	anass1rs 655	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ ℂ)
28	2	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐿 ∈ ℝ)
29	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑈 ∈ ℝ)
30		3factsumint1.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))
31	1	oveq1i 7356	. . . . . . . . 9 ⊢ (𝐴–cn→ℂ) = ((𝐿[,]𝑈)–cn→ℂ)
32	31	eleq2i 2823	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
33	30, 32	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
34		cnicciblnc 25771	. . . . . . 7 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
35	28, 29, 33, 34	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
36	10, 27, 35	iblmulc2 25759	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹)
37	31	eleq2i 2823	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
38	23, 37	sylib 218	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
39		cniccbdd 25389	. . . . . . . 8 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
40	2, 3, 38, 39	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
41	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
42	8	ralrimiva 3124	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
43		dmmptg 6189	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = 𝐴)
44	42, 43	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = 𝐴)
45	44, 1	eqtrdi 2782	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝐿[,]𝑈))
46	45	raleqdv 3292	. . . . . . . 8 ⊢ (𝜑 → (∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞 ↔ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞))
47	46	rexbidv 3156	. . . . . . 7 ⊢ (𝜑 → (∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞 ↔ ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞))
48	47	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞 ↔ ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞))
49	41, 48	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
50		bddmulibl 25767	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹ ∧ ∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
51	26, 36, 49, 50	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
52	22, 51	eqeltrrd 2832	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹)
53	6, 7, 14, 52	itgfsum 25755	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹ ∧ ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥))
54	53	simprd 495	1 ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 Vcvv 3436 class class class wbr 5089 ↦ cmpt 5170 dom cdm 5614 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 Fincfn 8869 ℂcc 11004 ℝcr 11005 · cmul 11011 ≤ cle 11147 [,]cicc 13248 abscabs 15141 Σcsu 15593 –cn→ccncf 24796 volcvol 25391 MblFncmbf 25542 𝐿¹cibl 25545 ∫citg 25546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cc 10326 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-omul 8390 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cn 23142 df-cnp 23143 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 df-itg2 25549 df-ibl 25550 df-itg 25551 df-0p 25598

This theorem is referenced by: 3factsumint 42117

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