lsmhash - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lsmhash		Structured version Visualization version GIF version

Theorem lsmhash 19614

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 7446	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2730	. . . 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 2730	. . . . . . 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 2730	. . . . . . 7 ⊢ (proj₁‘𝐺) = (proj₁‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19606	. . . . . 6 ⊢ (𝜑 → (𝑇(proj₁‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj₁‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19607	. . . . . 6 ⊢ (𝜑 → (𝑈(proj₁‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelcdmda 7085	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj₁‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5714	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj₁‘𝐺)𝑈)‘𝑥), ((𝑈(proj₁‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
17	7, 8	jca 510	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18		xp1st 8009	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1^st ‘𝑦) ∈ 𝑇)
19		xp2nd 8010	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2^nd ‘𝑦) ∈ 𝑈)
20	18, 19	jca 510	. . . . 5 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1^st ‘𝑦) ∈ 𝑇 ∧ (2^nd ‘𝑦) ∈ 𝑈))
21	3, 4	lsmelvali 19559	. . . . 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 594	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1^st ‘𝑦)(+_g‘𝐺)(2^nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
23	7	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
24	8	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
25	9	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
26	10	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
27		simprl 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
28	18	ad2antll 725	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1^st ‘𝑦) ∈ 𝑇)
29	19	ad2antll 725	. . . . . . 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 19609	. . . . . 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 2737	. . . . . . 7 ⊢ (((𝑇(proj₁‘𝐺)𝑈)‘𝑥) = (1^st ‘𝑦) ↔ (1^st ‘𝑦) = ((𝑇(proj₁‘𝐺)𝑈)‘𝑥))
32		eqcom 2737	. . . . . . 7 ⊢ (((𝑈(proj₁‘𝐺)𝑇)‘𝑥) = (2^nd ‘𝑦) ↔ (2^nd ‘𝑦) = ((𝑈(proj₁‘𝐺)𝑇)‘𝑥))
33	31, 32	anbi12i 625	. . . . . 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 286	. . . . 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 8019	. . . . . 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 725	. . . . 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 281	. . . 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 7662	. . 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 14317	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14398	. . 3 ⊢ ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
43	40, 41, 42	syl2anc 582	. 2 ⊢ (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
44	39, 43	eqtrd 2770	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 {csn 4627 ⟨cop 4633 ↦ cmpt 5230 × cxp 5673 ‘cfv 6542 (class class class)co 7411 1^st c1st 7975 2^nd c2nd 7976 Fincfn 8941 · cmul 11117 ♯chash 14294 +_gcplusg 17201 0_gc0g 17389 SubGrpcsubg 19036 Cntzccntz 19220 LSSumclsm 19543 proj₁cpj1 19544

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-hash 14295 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cntz 19222 df-lsm 19545 df-pj1 19546

This theorem is referenced by: ablfacrp2 19978 ablfac1eulem 19983 ablfac1eu 19984 pgpfaclem2 19993

W3C validator