Theorem lsmhash 19680

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 7402	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
2		eqid 2736	. . . 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 2736	. . . . . . 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 2736	. . . . . . 7 ⊢ (proj1‘𝐺) = (proj1‘𝐺)
12	3, 4, 5, 6, 7, 8, 9, 10, 11	pj1f 19672	. . . . . 6 ⊢ (𝜑 → (𝑇(proj1‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
13	12	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
14	3, 4, 5, 6, 7, 8, 9, 10, 11	pj2f 19673	. . . . . 6 ⊢ (𝜑 → (𝑈(proj1‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
15	14	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
16	13, 15	opelxpd 5670	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
17	7, 8	jca 511	. . . . 5 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18		xp1st 7974	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1st ‘𝑦) ∈ 𝑇)
19		xp2nd 7975	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2nd ‘𝑦) ∈ 𝑈)
20	18, 19	jca 511	. . . . 5 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈))
21	3, 4	lsmelvali 19625	. . . . 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 597	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
23	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
24	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
25	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
26	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
27		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
28	18	ad2antll 730	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st ‘𝑦) ∈ 𝑇)
29	19	ad2antll 730	. . . . . . 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 19675	. . . . . 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 2743	. . . . . . 7 ⊢ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥))
32		eqcom 2743	. . . . . . 7 ⊢ (((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦) ↔ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))
33	31, 32	anbi12i 629	. . . . . 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 7984	. . . . . 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 730	. . . . 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 282	. . . 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 7621	. . 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 14315	. 2 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = (♯‘(𝑇 × 𝑈)))
40		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
41		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
42		hashxp 14396	. . 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 2771	1 ⊢ (𝜑 → (♯‘(𝑇 ⊕ 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  {csn 4567  ⟨cop 4573   ↦ cmpt 5166   × cxp 5629  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Fincfn 8893   · cmul 11043  ♯chash 14292  +_gcplusg 17220  0_gc0g 17402  SubGrpcsubg 19096  Cntzccntz 19290  LSSumclsm 19609  proj₁cpj1 19610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-cntz 19292  df-lsm 19611  df-pj1 19612

This theorem is referenced by:  ablfacrp2  20044  ablfac1eulem  20049  ablfac1eu  20050  pgpfaclem2  20059