3factsumint1 - Mathbox for metakunt

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Theorem 3factsumint1 42003

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 25615	. . . . 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 2843	. . 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 11279	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐺 · 𝐻) ∈ ℂ)
14	9, 13	mulcld 11279	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐹 · (𝐺 · 𝐻)) ∈ ℂ)
15		ovex 7464	. . . . . . 7 ⊢ (𝐿[,]𝑈) ∈ V
16	1, 15	eqeltri 2835	. . . . . 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 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹))
21		eqidd 2736	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) = (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)))
22	17, 18, 19, 20, 21	offval2 7717	. . . 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 25708	. . . . . . 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 7441	. . . . . . . . 9 ⊢ (𝐴–cn→ℂ) = ((𝐿[,]𝑈)–cn→ℂ)
32	31	eleq2i 2831	. . . . . . . 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 25893	. . . . . . 7 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
35	28, 29, 33, 34	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
36	10, 27, 35	iblmulc2 25881	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹)
37	31	eleq2i 2831	. . . . . . . . 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 25510	. . . . . . . 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 1370	. . . . . . 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 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
43		dmmptg 6264	. . . . . . . . . . 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 2791	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝐿[,]𝑈))
46	45	raleqdv 3324	. . . . . . . 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 3177	. . . . . . 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 25889	. . . . 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 1370	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
52	22, 51	eqeltrrd 2840	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹)
53	6, 7, 14, 52	itgfsum 25877	. 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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ∘_f cof 7695 Fincfn 8984 ℂcc 11151 ℝcr 11152 · cmul 11158 ≤ cle 11294 [,]cicc 13387 abscabs 15270 Σcsu 15719 –cn→ccncf 24916 volcvol 25512 MblFncmbf 25663 𝐿¹cibl 25666 ∫citg 25667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cn 23251 df-cnp 23252 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719

This theorem is referenced by: 3factsumint 42007

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