3factsumint1 - Mathbox for metakunt

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Theorem 3factsumint1 42816

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 25736	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐿[,]𝑈) ∈ dom vol)
5	2, 3, 4	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐿[,]𝑈) ∈ dom vol)
6	1, 5	eqeltrid 2866	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
7		3factsumint1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		3factsumint1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
9	8	adantrr 729	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐹 ∈ ℂ)
10		3factsumint1.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
11	10	adantrl 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐺 ∈ ℂ)
12		3factsumint1.8	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
13	11, 12	mulcld 11235	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐺 · 𝐻) ∈ ℂ)
14	9, 13	mulcld 11235	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐹 · (𝐺 · 𝐻)) ∈ ℂ)
15		ovex 7445	. . . . . . 7 ⊢ (𝐿[,]𝑈) ∈ V
16	1, 15	eqeltri 2858	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ∈ V)
18	9	anass1rs 667	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
19	13	anass1rs 667	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 · 𝐻) ∈ ℂ)
20		eqidd 2763	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹))
21		eqidd 2763	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) = (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)))
22	17, 18, 19, 20, 21	offval2 7696	. . . 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 25829	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
25	6, 23, 24	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
26	25	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
27	12	anass1rs 667	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ ℂ)
28	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐿 ∈ ℝ)
29	3	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑈 ∈ ℝ)
30		3factsumint1.9	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))
31	1	oveq1i 7422	. . . . . . . . 9 ⊢ (𝐴–cn→ℂ) = ((𝐿[,]𝑈)–cn→ℂ)
32	31	eleq2i 2854	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
33	30, 32	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ))
34		cnicciblnc 26013	. . . . . . 7 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
35	28, 29, 33, 34	syl3anc 1397	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
36	10, 27, 35	iblmulc2 26001	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹)
37	31	eleq2i 2854	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
38	23, 37	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ ((𝐿[,]𝑈)–cn→ℂ))
39		cniccbdd 25631	. . . . . . . 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 1397	. . . . . . 7 ⊢ (𝜑 → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
41	40	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ (𝐿[,]𝑈)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
42	8	ralrimiva 3156	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
43		dmmptg 6242	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = 𝐴)
44	42, 43	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = 𝐴)
45	44, 1	eqtrdi 2813	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝐿[,]𝑈))
46	45	raleqdv 3322	. . . . . . . 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 3188	. . . . . . 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 485	. . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∃𝑞 ∈ ℝ ∀𝑟 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐹)(abs‘((𝑥 ∈ 𝐴 ↦ 𝐹)‘𝑟)) ≤ 𝑞)
50		bddmulibl 26009	. . . . 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 1397	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
52	22, 51	eqeltrrd 2863	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹)
53	6, 7, 14, 52	itgfsum 25997	. 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 500	1 ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 ∃wrex 3088 Vcvv 3454 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5660 ‘cfv 6536 (class class class)co 7412 ∘_f cof 7674 Fincfn 8941 ℂcc 11104 ℝcr 11105 · cmul 11111 ≤ cle 11250 [,]cicc 13381 abscabs 15292 Σcsu 15744 –cn→ccncf 25046 volcvol 25633 MblFncmbf 25784 𝐿¹cibl 25787 ∫citg 25788

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cc 10425 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 proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9894 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-hash 14374 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cn 23395 df-cnp 23396 df-cmp 23555 df-tx 23730 df-hmeo 23923 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-ovol 25634 df-vol 25635 df-mbf 25789 df-itg1 25790 df-itg2 25791 df-ibl 25792 df-itg 25793 df-0p 25840

This theorem is used by: 3factsumint 42820

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