Theorem lsmhash 19311

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 = (0g‘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 7310	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2738	. . . 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 2738	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		lsmhash.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝐺)
5		lsmhash.o	. . . . . . 7 ⊢ 0 = (0g‘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 2738	. . . . . . 7 ⊢ (proj1‘𝐺) = (proj1‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19303	. . . . . 6 ⊢ (𝜑 → (𝑇(proj1‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19304	. . . . . 6 ⊢ (𝜑 → (𝑈(proj1‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5627	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
17	7, 8	jca 512	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18		xp1st 7863	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1st ‘𝑦) ∈ 𝑇)
19		xp2nd 7864	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2nd ‘𝑦) ∈ 𝑈)
20	18, 19	jca 512	. . . . 5 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈))
21	3, 4	lsmelvali 19255	. . . . 5 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
22	17, 20, 21	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
23	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
24	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
25	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
26	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
27		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
28	18	ad2antll 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st ‘𝑦) ∈ 𝑇)
29	19	ad2antll 726	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd ‘𝑦) ∈ 𝑈)
30	3, 4, 5, 6, 23, 24, 25, 26, 11, 27, 28, 29	pj1eq 19306	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ↔ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ∧ ((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦))))
31		eqcom 2745	. . . . . . 7 ⊢ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥))
32		eqcom 2745	. . . . . . 7 ⊢ (((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦) ↔ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))
33	31, 32	anbi12i 627	. . . . . 6 ⊢ ((((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ∧ ((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦)) ↔ ((1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∧ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥)))
34	30, 33	bitrdi 287	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ↔ ((1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∧ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))))
35		eqop 7873	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ↔ ((1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∧ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))))
36	35	ad2antll 726	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑦 = ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ↔ ((1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∧ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))))
37	34, 36	bitr4d 281	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩))
38	2, 16, 22, 37	f1o2d 7523	. . 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 14068	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14149	. . 3 ⊢ ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
43	40, 41, 42	syl2anc 584	. 2 ⊢ (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
44	39, 43	eqtrd 2778	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  {csn 4561  ⟨cop 4567   ↦ cmpt 5157   × cxp 5587  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Fincfn 8733   · cmul 10876  ♯chash 14044  +_gcplusg 16962  0_gc0g 17150  SubGrpcsubg 18749  Cntzccntz 18921  LSSumclsm 19239  proj₁cpj1 19240

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-cntz 18923  df-lsm 19241  df-pj1 19242

This theorem is referenced by:  ablfacrp2  19670  ablfac1eulem  19675  ablfac1eu  19676  pgpfaclem2  19685