3factsumint - Mathbox for metakunt

	Mathbox for metakunt		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 3factsumint		Structured version Visualization version GIF version

Theorem 3factsumint 42482

Description: Helpful equation for lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.)

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

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

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

**Proof of Theorem 3factsumint**
Step	Hyp	Ref	Expression
1		3factsumint.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ Fin)
2		3factsumint.5	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))
3		cncff 24874	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶ℂ)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶ℂ)
5		eqid 2737	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐹)
6	5	fmpt 7058	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐹):𝐴⟶ℂ)
7	4, 6	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐹 ∈ ℂ)
8	7	r19.21bi 3230	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)
9		3factsumint.6	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)
10		3factsumint.7	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))
11		cncff 24874	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐻):𝐴⟶ℂ)
12	10, 11	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻):𝐴⟶ℂ)
13		eqid 2737	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐻) = (𝑥 ∈ 𝐴 ↦ 𝐻)
14	13	fmpt 7058	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐻 ∈ ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐻):𝐴⟶ℂ)
15	12, 14	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐻 ∈ ℂ)
16	15	r19.21bi 3230	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ ℂ)
17		anass 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)))
18		ancom 460	. . . . . . . 8 ⊢ ((𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))
19	18	anbi2i 624	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) ↔ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
20	17, 19	bitri 275	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)))
21	20	imbi1i 349	. . . . 5 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐻 ∈ ℂ) ↔ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ))
22	16, 21	mpbi 230	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)
23	1, 8, 9, 22	3factsumint4 42481	. . 3 ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥)
24		3factsumint.1	. . . 4 ⊢ 𝐴 = (𝐿[,]𝑈)
25		3factsumint.3	. . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ)
26		3factsumint.4	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ)
27	24, 1, 25, 26, 8, 2, 9, 22, 10	3factsumint1 42478	. . 3 ⊢ (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)
28	23, 27	eqtr3d 2774	. 2 ⊢ (𝜑 → ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)
29	8, 9, 22	3factsumint2 42479	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥)
30	24, 25, 26, 8, 2, 9, 22, 10	3factsumint3 42480	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥))
31	28, 29, 30	3eqtrd 2776	1 ⊢ (𝜑 → ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ↦ cmpt 5167 ⟶wf 6490 (class class class)co 7362 Fincfn 8888 ℂcc 11031 ℝcr 11032 · cmul 11038 [,]cicc 13296 Σcsu 15643 –cn→ccncf 24857 ∫citg 25599

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-inf2 9557 ax-cc 10352 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

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 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-se 5580 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7626 df-ofr 7627 df-om 7813 df-1st 7937 df-2nd 7938 df-supp 8106 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-oadd 8404 df-omul 8405 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-fi 9319 df-sup 9350 df-inf 9351 df-oi 9420 df-dju 9820 df-card 9858 df-acn 9861 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cn 23206 df-cnp 23207 df-cmp 23366 df-tx 23541 df-hmeo 23734 df-xms 24299 df-ms 24300 df-tms 24301 df-cncf 24859 df-ovol 25445 df-vol 25446 df-mbf 25600 df-itg1 25601 df-itg2 25602 df-ibl 25603 df-itg 25604 df-0p 25651

This theorem is referenced by: lcmineqlem2 42487

W3C validator