Theorem lsmhash 19928

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 7459	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2760	. . . 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 2760	. . . . . . 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 2760	. . . . . . 7 ⊢ (proj1‘𝐺) = (proj1‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19920	. . . . . 6 ⊢ (𝜑 → (𝑇(proj1‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelcdmda 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19921	. . . . . 6 ⊢ (𝜑 → (𝑈(proj1‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelcdmda 7089	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5690	. . . 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 8037	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1st ‘𝑦) ∈ 𝑇)
19		xp2nd 8038	. . . . . 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 19873	. . . . 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 19923	. . . . . 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 2767	. . . . . . 7 ⊢ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥))
32		eqcom 2767	. . . . . . 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 8047	. . . . . 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 7679	. . 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 14504	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14586	. . 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 2795	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088  {csn 4584  ⟨cop 4834   ↦ cmpt 5185   × cxp 5649  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004  Fincfn 8981   · cmul 11212  ♯chash 14481  +_gcplusg 17437  0_gc0g 17619  SubGrpcsubg 19339  Cntzccntz 19538  LSSumclsm 19857  proj₁cpj1 19858

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-oadd 8488  df-er 8725  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-n0 12614  df-z 12701  df-uz 12973  df-fz 13647  df-hash 14482  df-sets 17351  df-slot 17369  df-ndx 17381  df-base 17397  df-ress 17418  df-plusg 17450  df-0g 17621  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-minusg 19157  df-sbg 19158  df-subg 19342  df-cntz 19540  df-lsm 19859  df-pj1 19860

This theorem is used by:  ablfacrp2  20292  ablfac1eulem  20297  ablfac1eu  20298  pgpfaclem2  20307