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Theorem lsmhash 19573

Description: The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
lsmhash.p	⊢ ⊕ = (LSSum‘𝐺)
lsmhash.o	⊢ 0 = (0_g‘𝐺)
lsmhash.z	⊢ 𝑍 = (Cntz‘𝐺)
lsmhash.t	⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
lsmhash.u	⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
lsmhash.i	⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
lsmhash.s	⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
lsmhash.1	⊢ (𝜑 → 𝑇 ∈ Fin)
lsmhash.2	⊢ (𝜑 → 𝑈 ∈ Fin)

**Assertion**
Ref	Expression
lsmhash	⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

Proof of Theorem lsmhash Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovexd 7444	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2733	. . . 4 ⊢ (𝑥 ∈ (𝑇 ⊕ 𝑈) ↦ ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩) = (𝑥 ∈ (𝑇 ⊕ 𝑈) ↦ ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩)
3		eqid 2733	. . . . . . 7 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
4		lsmhash.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
5		lsmhash.o	. . . . . . 7 ⊢ 0 = (0_g‘𝐺)
6		lsmhash.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝐺)
7		lsmhash.t	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
8		lsmhash.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
9		lsmhash.i	. . . . . . 7 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
10		lsmhash.s	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
11		eqid 2733	. . . . . . 7 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19565	. . . . . 6 ⊢ (𝜑 → (𝑇(proj₁‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelcdmda 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19566	. . . . . 6 ⊢ (𝜑 → (𝑈(proj₁‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelcdmda 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj₁‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5716	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
17	7, 8	jca 513	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18		xp1st 8007	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1^st ‘𝑦) ∈ 𝑇)
19		xp2nd 8008	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2^nd ‘𝑦) ∈ 𝑈)
20	18, 19	jca 513	. . . . 5 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1^st ‘𝑦) ∈ 𝑇 ∧ (2^nd ‘𝑦) ∈ 𝑈))
21	3, 4	lsmelvali 19518	. . . . 5 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1^st ‘𝑦) ∈ 𝑇 ∧ (2^nd ‘𝑦) ∈ 𝑈)) → ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
22	17, 20, 21	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
23	7	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
24	8	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
25	9	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
26	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
27		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
28	18	ad2antll 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1^st ‘𝑦) ∈ 𝑇)
29	19	ad2antll 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2^nd ‘𝑦) ∈ 𝑈)
30	3, 4, 5, 6, 23, 24, 25, 26, 11, 27, 28, 29	pj1eq 19568	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ↔ (((𝑇(proj₁‘𝐺)𝑈)‘𝑥) = (1^st ‘𝑦) ∧ ((𝑈(proj₁‘𝐺)𝑇)‘𝑥) = (2^nd ‘𝑦))))
31		eqcom 2740	. . . . . . 7 ⊢ (((𝑇(proj₁‘𝐺)𝑈)‘𝑥) = (1^st ‘𝑦) ↔ (1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥))
32		eqcom 2740	. . . . . . 7 ⊢ (((𝑈(proj₁‘𝐺)𝑇)‘𝑥) = (2^nd ‘𝑦) ↔ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥))
33	31, 32	anbi12i 628	. . . . . 6 ⊢ ((((𝑇(proj₁‘𝐺)𝑈)‘𝑥) = (1^st ‘𝑦) ∧ ((𝑈(proj₁‘𝐺)𝑇)‘𝑥) = (2^nd ‘𝑦)) ↔ ((1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∧ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)))
34	30, 33	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ↔ ((1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∧ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥))))
35		eqop 8017	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩ ↔ ((1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∧ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥))))
36	35	ad2antll 728	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑦 = ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩ ↔ ((1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∧ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥))))
37	34, 36	bitr4d 282	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ↔ 𝑦 = ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩))
38	2, 16, 22, 37	f1o2d 7660	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↦ ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩):(𝑇 ⊕ 𝑈)–1-1-onto→(𝑇 × 𝑈))
39	1, 38	hasheqf1od 14313	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14394	. . 3 ⊢ ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
43	40, 41, 42	syl2anc 585	. 2 ⊢ (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
44	39, 43	eqtrd 2773	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 {csn 4629 ⟨cop 4635 ↦ cmpt 5232 × cxp 5675 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 Fincfn 8939 · cmul 11115 ♯chash 14290 +_gcplusg 17197 0_gc0g 17385 SubGrpcsubg 19000 Cntzccntz 19179 LSSumclsm 19502 proj₁cpj1 19503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-cntz 19181 df-lsm 19504 df-pj1 19505

This theorem is referenced by: ablfacrp2 19937 ablfac1eulem 19942 ablfac1eu 19943 pgpfaclem2 19952

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