Theorem lsmhash 17941

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		ovex 6577	. . . . 5 ⊢ (𝑇 ⊕ 𝑈) ∈ V
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ V)
3		lsmhash.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺))
4		lsmhash.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺))
5		xpexg 6858	. . . . 5 ⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 × 𝑈) ∈ V)
6	3, 4, 5	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑇 × 𝑈) ∈ V)
7		eqid 2610	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		lsmhash.p	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝐺)
9		lsmhash.o	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
10		lsmhash.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝐺)
11		lsmhash.i	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 })
12		lsmhash.s	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ (𝑍‘𝑈))
13		eqid 2610	. . . . . . . 8 ⊢ (proj1‘𝐺) = (proj1‘𝐺)
14	7, 8, 9, 10, 3, 4, 11, 12, 13	pj1f 17933	. . . . . . 7 ⊢ (𝜑 → (𝑇(proj1‘𝐺)𝑈):(𝑇 ⊕ 𝑈)⟶𝑇)
15	14	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇)
16	7, 8, 9, 10, 3, 4, 11, 12, 13	pj2f 17934	. . . . . . 7 ⊢ (𝜑 → (𝑈(proj1‘𝐺)𝑇):(𝑇 ⊕ 𝑈)⟶𝑈)
17	16	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈)
18		opelxpi 5072	. . . . . 6 ⊢ ((((𝑇(proj1‘𝐺)𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑈(proj1‘𝐺)𝑇)‘𝑥) ∈ 𝑈) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
19	15, 17, 18	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ⊕ 𝑈)) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
20	19	ex 449	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ 𝑈) → ⟨((𝑇(proj1‘𝐺)𝑈)‘𝑥), ((𝑈(proj1‘𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈)))
21	3, 4	jca 553	. . . . . 6 ⊢ (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
22		xp1st 7089	. . . . . . 7 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (1st ‘𝑦) ∈ 𝑇)
23		xp2nd 7090	. . . . . . 7 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → (2nd ‘𝑦) ∈ 𝑈)
24	22, 23	jca 553	. . . . . 6 ⊢ (𝑦 ∈ (𝑇 × 𝑈) → ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈))
25	7, 8	lsmelvali 17888	. . . . . 6 ⊢ (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st ‘𝑦) ∈ 𝑇 ∧ (2nd ‘𝑦) ∈ 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
26	21, 24, 25	syl2an 493	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑇 × 𝑈)) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈))
27	26	ex 449	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑇 × 𝑈) → ((1st ‘𝑦)(+g‘𝐺)(2nd ‘𝑦)) ∈ (𝑇 ⊕ 𝑈)))
28	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
29	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
30	11	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇 ∩ 𝑈) = { 0 })
31	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
32		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 ⊕ 𝑈))
33	22	ad2antll 761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st ‘𝑦) ∈ 𝑇)
34	23	ad2antll 761	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd ‘𝑦) ∈ 𝑈)
35	7, 8, 9, 10, 28, 29, 30, 31, 13, 32, 33, 34	pj1eq 17936	. . . . . . 7 ⊢ ((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))))
36		eqcom 2617	. . . . . . . 8 ⊢ (((𝑇(proj1‘𝐺)𝑈)‘𝑥) = (1st ‘𝑦) ↔ (1st ‘𝑦) = ((𝑇(proj1‘𝐺)𝑈)‘𝑥))
37		eqcom 2617	. . . . . . . 8 ⊢ (((𝑈(proj1‘𝐺)𝑇)‘𝑥) = (2nd ‘𝑦) ↔ (2nd ‘𝑦) = ((𝑈(proj1‘𝐺)𝑇)‘𝑥))
38	36, 37	anbi12i 729	. . . . . . 7 ⊢ ((((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)))
39	35, 38	syl6bb 275	. . . . . 6 ⊢ ((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))))
40		eqop 7099	. . . . . . 7 ⊢ (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))))
41	40	ad2antll 761	. . . . . 6 ⊢ ((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))))
42	39, 41	bitr4d 270	. . . . 5 ⊢ ((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)⟩))
43	42	ex 449	. . . 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)⟩)))
44	2, 6, 20, 27, 43	en3d 7878	. . 3 ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ≈ (𝑇 × 𝑈))
45		hasheni 12998	. . 3 ⊢ ((𝑇 ⊕ 𝑈) ≈ (𝑇 × 𝑈) → (#‘(𝑇 ⊕ 𝑈)) = (#‘(𝑇 × 𝑈)))
46	44, 45	syl 17	. 2 ⊢ (𝜑 → (#‘(𝑇 ⊕ 𝑈)) = (#‘(𝑇 × 𝑈)))
47		lsmhash.1	. . 3 ⊢ (𝜑 → 𝑇 ∈ Fin)
48		lsmhash.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ Fin)
49		hashxp 13081	. . 3 ⊢ ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
50	47, 48, 49	syl2anc 691	. 2 ⊢ (𝜑 → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
51	46, 50	eqtrd 2644	1 ⊢ (𝜑 → (#‘(𝑇 ⊕ 𝑈)) = ((#‘𝑇) · (#‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ≈ cen 7838  Fincfn 7841   · cmul 9820  #chash 12979  +_gcplusg 15768  0_gc0g 15923  SubGrpcsubg 17411  Cntzccntz 17571  LSSumclsm 17872  proj₁cpj1 17873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-lsm 17874  df-pj1 17875

This theorem is referenced by:  ablfacrp2  18289  ablfac1eulem  18294  ablfac1eu  18295  pgpfaclem2  18304