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Theorem lsmhash 18039
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 = (0g𝐺)
lsmhash.z 𝑍 = (Cntz‘𝐺)
lsmhash.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
lsmhash.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
lsmhash.i (𝜑 → (𝑇𝑈) = { 0 })
lsmhash.s (𝜑𝑇 ⊆ (𝑍𝑈))
lsmhash.1 (𝜑𝑇 ∈ Fin)
lsmhash.2 (𝜑𝑈 ∈ 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.
StepHypRef Expression
1 ovex 6632 . . . . 5 (𝑇 𝑈) ∈ V
21a1i 11 . . . 4 (𝜑 → (𝑇 𝑈) ∈ V)
3 lsmhash.t . . . . 5 (𝜑𝑇 ∈ (SubGrp‘𝐺))
4 lsmhash.u . . . . 5 (𝜑𝑈 ∈ (SubGrp‘𝐺))
5 xpexg 6913 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 × 𝑈) ∈ V)
63, 4, 5syl2anc 692 . . . 4 (𝜑 → (𝑇 × 𝑈) ∈ V)
7 eqid 2621 . . . . . . . 8 (+g𝐺) = (+g𝐺)
8 lsmhash.p . . . . . . . 8 = (LSSum‘𝐺)
9 lsmhash.o . . . . . . . 8 0 = (0g𝐺)
10 lsmhash.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
11 lsmhash.i . . . . . . . 8 (𝜑 → (𝑇𝑈) = { 0 })
12 lsmhash.s . . . . . . . 8 (𝜑𝑇 ⊆ (𝑍𝑈))
13 eqid 2621 . . . . . . . 8 (proj1𝐺) = (proj1𝐺)
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 18031 . . . . . . 7 (𝜑 → (𝑇(proj1𝐺)𝑈):(𝑇 𝑈)⟶𝑇)
1514ffvelrnda 6315 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇)
167, 8, 9, 10, 3, 4, 11, 12, 13pj2f 18032 . . . . . . 7 (𝜑 → (𝑈(proj1𝐺)𝑇):(𝑇 𝑈)⟶𝑈)
1716ffvelrnda 6315 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈)
18 opelxpi 5108 . . . . . 6 ((((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
1915, 17, 18syl2anc 692 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
2019ex 450 . . . 4 (𝜑 → (𝑥 ∈ (𝑇 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈)))
213, 4jca 554 . . . . . 6 (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
22 xp1st 7143 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (1st𝑦) ∈ 𝑇)
23 xp2nd 7144 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (2nd𝑦) ∈ 𝑈)
2422, 23jca 554 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈))
257, 8lsmelvali 17986 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2621, 24, 25syl2an 494 . . . . 5 ((𝜑𝑦 ∈ (𝑇 × 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2726ex 450 . . . 4 (𝜑 → (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈)))
283adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
294adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
3011adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇𝑈) = { 0 })
3112adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
32 simprl 793 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 𝑈))
3322ad2antll 764 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st𝑦) ∈ 𝑇)
3423ad2antll 764 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd𝑦) ∈ 𝑈)
357, 8, 9, 10, 28, 29, 30, 31, 13, 32, 33, 34pj1eq 18034 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦))))
36 eqcom 2628 . . . . . . . 8 (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ↔ (1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥))
37 eqcom 2628 . . . . . . . 8 (((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦) ↔ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))
3836, 37anbi12i 732 . . . . . . 7 ((((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥)))
3935, 38syl6bb 276 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
40 eqop 7153 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
4140ad2antll 764 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
4239, 41bitr4d 271 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩))
4342ex 450 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈)) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩)))
442, 6, 20, 27, 43en3d 7936 . . 3 (𝜑 → (𝑇 𝑈) ≈ (𝑇 × 𝑈))
45 hasheni 13076 . . 3 ((𝑇 𝑈) ≈ (𝑇 × 𝑈) → (#‘(𝑇 𝑈)) = (#‘(𝑇 × 𝑈)))
4644, 45syl 17 . 2 (𝜑 → (#‘(𝑇 𝑈)) = (#‘(𝑇 × 𝑈)))
47 lsmhash.1 . . 3 (𝜑𝑇 ∈ Fin)
48 lsmhash.2 . . 3 (𝜑𝑈 ∈ Fin)
49 hashxp 13161 . . 3 ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
5047, 48, 49syl2anc 692 . 2 (𝜑 → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
5146, 50eqtrd 2655 1 (𝜑 → (#‘(𝑇 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555  {csn 4148  cop 4154   class class class wbr 4613   × cxp 5072  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  cen 7896  Fincfn 7899   · cmul 9885  #chash 13057  +gcplusg 15862  0gc0g 16021  SubGrpcsubg 17509  Cntzccntz 17669  LSSumclsm 17970  proj1cpj1 17971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-cntz 17671  df-lsm 17972  df-pj1 17973
This theorem is referenced by:  ablfacrp2  18387  ablfac1eulem  18392  ablfac1eu  18393  pgpfaclem2  18402
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