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Theorem 3factsumint1 39173

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 24149	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐿[,]𝑈) ∈ dom vol)
5	2, 3, 4	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐿[,]𝑈) ∈ dom vol)
6	1, 5	eqeltrid 2915	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
7		3factsumint1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		3factsumint1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
9	8	adantrr 715	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐹 ∈ ℂ)
10		3factsumint1.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
11	10	adantrl 714	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐺 ∈ ℂ)
12		3factsumint1.8	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
13	11, 12	mulcld 10639	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐺 · 𝐻) ∈ ℂ)
14	9, 13	mulcld 10639	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐹 · (𝐺 · 𝐻)) ∈ ℂ)
15		ovex 7166	. . . . . . 7 ⊢ (𝐿[,]𝑈) ∈ V
16	1, 15	eqeltri 2907	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ∈ V)
18	9	anass1rs 653	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
19	13	anass1rs 653	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 · 𝐻) ∈ ℂ)
20		eqidd 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹))
21		eqidd 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) = (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)))
22	17, 18, 19, 20, 21	offval2 7404	. . . 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 24242	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
25	6, 23, 24	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
26	25	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
27	12	anass1rs 653	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ ℂ)
28	2	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐿 ∈ ℝ)
29	3	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑈 ∈ ℝ)
30		3factsumint1.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))
31	1	oveq1i 7143	. . . . . . . . 9 ⊢ (𝐴–cn→ℂ) = ((𝐿[,]𝑈)–cn→ℂ)
32	31	eleq2i 2902	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
33	30, 32	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
34		cnicciblnc 24425	. . . . . . 7 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
35	28, 29, 33, 34	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
36	10, 27, 35	iblmulc2 24413	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹)
37	31	eleq2i 2902	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
38	23, 37	sylib 220	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
39		cniccbdd 24044	. . . . . . . 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 1367	. . . . . . 7 ⊢ (𝜑 → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
41	40	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
42	8	ralrimiva 3169	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
43		dmmptg 6072	. . . . . . . . . . 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	syl6eq 2871	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝐿[,]𝑈))
46	45	raleqdv 3398	. . . . . . . 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 3284	. . . . . . 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 483	. . . . . 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 259	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
50		bddmulibl 24421	. . . . 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 1367	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
52	22, 51	eqeltrrd 2912	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹)
53	6, 7, 14, 52	itgfsum 24409	. 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 498	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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3125 ∃wrex 3126 Vcvv 3473 class class class wbr 5042 ↦ cmpt 5122 dom cdm 5531 ‘cfv 6331 (class class class)co 7133 ∘_f cof 7385 Fincfn 8487 ℂcc 10513 ℝcr 10514 · cmul 10520 ≤ cle 10654 [,]cicc 12720 abscabs 14573 Σcsu 15022 –cn→ccncf 23460 volcvol 24046 MblFncmbf 24197 𝐿¹cibl 24200 ∫citg 24201

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5166 ax-sep 5179 ax-nul 5186 ax-pow 5242 ax-pr 5306 ax-un 7439 ax-inf2 9082 ax-cc 9835 ax-cnex 10571 ax-resscn 10572 ax-1cn 10573 ax-icn 10574 ax-addcl 10575 ax-addrcl 10576 ax-mulcl 10577 ax-mulrcl 10578 ax-mulcom 10579 ax-addass 10580 ax-mulass 10581 ax-distr 10582 ax-i2m1 10583 ax-1ne0 10584 ax-1rid 10585 ax-rnegex 10586 ax-rrecex 10587 ax-cnre 10588 ax-pre-lttri 10589 ax-pre-lttrn 10590 ax-pre-ltadd 10591 ax-pre-mulgt0 10592 ax-pre-sup 10593 ax-addf 10594 ax-mulf 10595

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3007 df-nel 3111 df-ral 3130 df-rex 3131 df-reu 3132 df-rmo 3133 df-rab 3134 df-v 3475 df-sbc 3753 df-csb 3861 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4270 df-if 4444 df-pw 4517 df-sn 4544 df-pr 4546 df-tp 4548 df-op 4550 df-uni 4815 df-int 4853 df-iun 4897 df-iin 4898 df-disj 5008 df-br 5043 df-opab 5105 df-mpt 5123 df-tr 5149 df-id 5436 df-eprel 5441 df-po 5450 df-so 5451 df-fr 5490 df-se 5491 df-we 5492 df-xp 5537 df-rel 5538 df-cnv 5539 df-co 5540 df-dm 5541 df-rn 5542 df-res 5543 df-ima 5544 df-pred 6124 df-ord 6170 df-on 6171 df-lim 6172 df-suc 6173 df-iota 6290 df-fun 6333 df-fn 6334 df-f 6335 df-f1 6336 df-fo 6337 df-f1o 6338 df-fv 6339 df-isom 6340 df-riota 7091 df-ov 7136 df-oprab 7137 df-mpo 7138 df-of 7387 df-ofr 7388 df-om 7559 df-1st 7667 df-2nd 7668 df-supp 7809 df-wrecs 7925 df-recs 7986 df-rdg 8024 df-1o 8080 df-2o 8081 df-oadd 8084 df-omul 8085 df-er 8267 df-map 8386 df-pm 8387 df-ixp 8440 df-en 8488 df-dom 8489 df-sdom 8490 df-fin 8491 df-fsupp 8812 df-fi 8853 df-sup 8884 df-inf 8885 df-oi 8952 df-dju 9308 df-card 9346 df-acn 9349 df-pnf 10655 df-mnf 10656 df-xr 10657 df-ltxr 10658 df-le 10659 df-sub 10850 df-neg 10851 df-div 11276 df-nn 11617 df-2 11679 df-3 11680 df-4 11681 df-5 11682 df-6 11683 df-7 11684 df-8 11685 df-9 11686 df-n0 11877 df-z 11961 df-dec 12078 df-uz 12223 df-q 12328 df-rp 12369 df-xneg 12486 df-xadd 12487 df-xmul 12488 df-ioo 12721 df-ioc 12722 df-ico 12723 df-icc 12724 df-fz 12877 df-fzo 13018 df-fl 13146 df-mod 13222 df-seq 13354 df-exp 13415 df-hash 13676 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-limsup 14808 df-clim 14825 df-rlim 14826 df-sum 15023 df-struct 16464 df-ndx 16465 df-slot 16466 df-base 16468 df-sets 16469 df-ress 16470 df-plusg 16557 df-mulr 16558 df-starv 16559 df-sca 16560 df-vsca 16561 df-ip 16562 df-tset 16563 df-ple 16564 df-ds 16566 df-unif 16567 df-hom 16568 df-cco 16569 df-rest 16675 df-topn 16676 df-0g 16694 df-gsum 16695 df-topgen 16696 df-pt 16697 df-prds 16700 df-xrs 16754 df-qtop 16759 df-imas 16760 df-xps 16762 df-mre 16836 df-mrc 16837 df-acs 16839 df-mgm 17831 df-sgrp 17880 df-mnd 17891 df-submnd 17936 df-mulg 18204 df-cntz 18426 df-cmn 18887 df-psmet 20513 df-xmet 20514 df-met 20515 df-bl 20516 df-mopn 20517 df-cnfld 20522 df-top 21478 df-topon 21495 df-topsp 21517 df-bases 21530 df-cn 21811 df-cnp 21812 df-cmp 21971 df-tx 22146 df-hmeo 22339 df-xms 22906 df-ms 22907 df-tms 22908 df-cncf 23462 df-ovol 24047 df-vol 24048 df-mbf 24202 df-itg1 24203 df-itg2 24204 df-ibl 24205 df-itg 24206 df-0p 24253

This theorem is referenced by: 3factsumint 39177

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