Theorem lsmhash 19970

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 7486	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2763	. . . 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 2763	. . . . . . 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 2763	. . . . . . 7 ⊢ (proj1‘𝐺) = (proj1‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19962	. . . . . 6 ⊢ (𝜑 → (𝑇(proj1‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelcdmda 7116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19963	. . . . . 6 ⊢ (𝜑 → (𝑈(proj1‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelcdmda 7116	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5697	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
17	7, 8	jca 521	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18		xp1st 8067	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1st ‘𝑦) ∈ 𝑇)
19		xp2nd 8068	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2nd ‘𝑦) ∈ 𝑈)
20	18, 19	jca 521	. . . . 5 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈))
21	3, 4	lsmelvali 19915	. . . . 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 608	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
23	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
24	8	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
25	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
26	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
27		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
28	18	ad2antll 742	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st ‘𝑦) ∈ 𝑇)
29	19	ad2antll 742	. . . . . . 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 19965	. . . . . 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 2770	. . . . . . 7 ⊢ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥))
32		eqcom 2770	. . . . . . 7 ⊢ (((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦) ↔ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))
33	31, 32	anbi12i 640	. . . . . 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 290	. . . . 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 8077	. . . . . 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 742	. . . . 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 285	. . . 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 7707	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↦ ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩):(𝑇 ⊕ 𝑈)⟶1-1onto(𝑇 × 𝑈))
39	1, 38	hasheqf1od 14538	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14620	. . 3 ⊢ ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
43	40, 41, 42	syl2anc 596	. 2 ⊢ (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
44	39, 43	eqtrd 2798	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  {csn 4588  ⟨cop 4840   ↦ cmpt 5192   × cxp 5656  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Fincfn 9011   · cmul 11244  ♯chash 14515  +_gcplusg 17472  0_gc0g 17660  SubGrpcsubg 19380  Cntzccntz 19580  LSSumclsm 19899  proj₁cpj1 19900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-oadd 8518  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-dju 10019  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-n0 12647  df-z 12734  df-uz 13006  df-fz 13680  df-hash 14516  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-grp 19197  df-minusg 19198  df-sbg 19199  df-subg 19383  df-cntz 19582  df-lsm 19901  df-pj1 19902

This theorem is used by:  ablfacrp2  20334  ablfac1eulem  20339  ablfac1eu  20340  pgpfaclem2  20349