3factsumint1 - Mathbox for metakunt

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Theorem 3factsumint1 42343

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 25527	. . . . 5 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐿[,]𝑈) ∈ dom vol)
5	2, 3, 4	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐿[,]𝑈) ∈ dom vol)
6	1, 5	eqeltrid 2841	. . 3 ⊢ (𝜑 → 𝐴 ∈ dom vol)
7		3factsumint1.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
8		3factsumint1.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
9	8	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐹 ∈ ℂ)
10		3factsumint1.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
11	10	adantrl 717	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐺 ∈ ℂ)
12		3factsumint1.8	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
13	11, 12	mulcld 11156	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐺 · 𝐻) ∈ ℂ)
14	9, 13	mulcld 11156	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → (𝐹 · (𝐺 · 𝐻)) ∈ ℂ)
15		ovex 7393	. . . . . . 7 ⊢ (𝐿[,]𝑈) ∈ V
16	1, 15	eqeltri 2833	. . . . . 6 ⊢ 𝐴 ∈ V
17	16	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐴 ∈ V)
18	9	anass1rs 656	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
19	13	anass1rs 656	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝐺 · 𝐻) ∈ ℂ)
20		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹))
21		eqidd 2738	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) = (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)))
22	17, 18, 19, 20, 21	offval2 7644	. . . 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 25620	. . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
25	6, 23, 24	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ MblFn)
27	12	anass1rs 656	. . . . . 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 7370	. . . . . . . . 9 ⊢ (𝐴–cn→ℂ) = ((𝐿[,]𝑈)–cn→ℂ)
32	31	eleq2i 2829	. . . . . . . 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 25804	. . . . . . 7 ⊢ ((𝐿 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ ((𝐿[,]𝑈)–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
35	28, 29, 33, 34	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ 𝐿¹)
36	10, 27, 35	iblmulc2 25792	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻)) ∈ 𝐿¹)
37	31	eleq2i 2829	. . . . . . . . 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 25422	. . . . . . . 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 1374	. . . . . . 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 3129	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
43		dmmptg 6201	. . . . . . . . . . 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 2788	. . . . . . . . 9 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝐿[,]𝑈))
46	45	raleqdv 3297	. . . . . . . 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 3161	. . . . . . 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 25800	. . . . 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 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ↦ 𝐹) ∘_f · (𝑥 ∈ 𝐴 ↦ (𝐺 · 𝐻))) ∈ 𝐿¹)
52	22, 51	eqeltrrd 2838	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ (𝐹 · (𝐺 · 𝐻))) ∈ 𝐿¹)
53	6, 7, 14, 52	itgfsum 25788	. 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 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3061 Vcvv 3441 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5625 ‘cfv 6493 (class class class)co 7360 ∘_f cof 7622 Fincfn 8887 ℂcc 11028 ℝcr 11029 · cmul 11035 ≤ cle 11171 [,]cicc 13268 abscabs 15161 Σcsu 15613 –cn→ccncf 24829 volcvol 25424 MblFncmbf 25575 𝐿¹cibl 25578 ∫citg 25579

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cn 23175 df-cnp 23176 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-xms 24268 df-ms 24269 df-tms 24270 df-cncf 24831 df-ovol 25425 df-vol 25426 df-mbf 25580 df-itg1 25581 df-itg2 25582 df-ibl 25583 df-itg 25584 df-0p 25631

This theorem is referenced by: 3factsumint 42347

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