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Theorem lsmhash 18760
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 ovexd 7180 . . . 4 (𝜑 → (𝑇 𝑈) ∈ V)
2 lsmhash.t . . . . 5 (𝜑𝑇 ∈ (SubGrp‘𝐺))
3 lsmhash.u . . . . 5 (𝜑𝑈 ∈ (SubGrp‘𝐺))
42, 3xpexd 7463 . . . 4 (𝜑 → (𝑇 × 𝑈) ∈ V)
5 eqid 2818 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmhash.p . . . . . . . 8 = (LSSum‘𝐺)
7 lsmhash.o . . . . . . . 8 0 = (0g𝐺)
8 lsmhash.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
9 lsmhash.i . . . . . . . 8 (𝜑 → (𝑇𝑈) = { 0 })
10 lsmhash.s . . . . . . . 8 (𝜑𝑇 ⊆ (𝑍𝑈))
11 eqid 2818 . . . . . . . 8 (proj1𝐺) = (proj1𝐺)
125, 6, 7, 8, 2, 3, 9, 10, 11pj1f 18752 . . . . . . 7 (𝜑 → (𝑇(proj1𝐺)𝑈):(𝑇 𝑈)⟶𝑇)
1312ffvelrnda 6843 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇)
145, 6, 7, 8, 2, 3, 9, 10, 11pj2f 18753 . . . . . . 7 (𝜑 → (𝑈(proj1𝐺)𝑇):(𝑇 𝑈)⟶𝑈)
1514ffvelrnda 6843 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈)
1613, 15opelxpd 5586 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
1716ex 413 . . . 4 (𝜑 → (𝑥 ∈ (𝑇 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈)))
182, 3jca 512 . . . . . 6 (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
19 xp1st 7710 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (1st𝑦) ∈ 𝑇)
20 xp2nd 7711 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (2nd𝑦) ∈ 𝑈)
2119, 20jca 512 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈))
225, 6lsmelvali 18704 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2318, 21, 22syl2an 595 . . . . 5 ((𝜑𝑦 ∈ (𝑇 × 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2423ex 413 . . . 4 (𝜑 → (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈)))
252adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
263adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
279adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇𝑈) = { 0 })
2810adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
29 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 𝑈))
3019ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st𝑦) ∈ 𝑇)
3120ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd𝑦) ∈ 𝑈)
325, 6, 7, 8, 25, 26, 27, 28, 11, 29, 30, 31pj1eq 18755 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦))))
33 eqcom 2825 . . . . . . . 8 (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ↔ (1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥))
34 eqcom 2825 . . . . . . . 8 (((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦) ↔ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))
3533, 34anbi12i 626 . . . . . . 7 ((((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥)))
3632, 35syl6bb 288 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
37 eqop 7720 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
3837ad2antll 725 . . . . . 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))))
3936, 38bitr4d 283 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩))
4039ex 413 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈)) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩)))
411, 4, 17, 24, 40en3d 8534 . . 3 (𝜑 → (𝑇 𝑈) ≈ (𝑇 × 𝑈))
42 hasheni 13696 . . 3 ((𝑇 𝑈) ≈ (𝑇 × 𝑈) → (♯‘(𝑇 𝑈)) = (♯‘(𝑇 × 𝑈)))
4341, 42syl 17 . 2 (𝜑 → (♯‘(𝑇 𝑈)) = (♯‘(𝑇 × 𝑈)))
44 lsmhash.1 . . 3 (𝜑𝑇 ∈ Fin)
45 lsmhash.2 . . 3 (𝜑𝑈 ∈ Fin)
46 hashxp 13783 . . 3 ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4744, 45, 46syl2anc 584 . 2 (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4843, 47eqtrd 2853 1 (𝜑 → (♯‘(𝑇 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  cin 3932  wss 3933  {csn 4557  cop 4563   class class class wbr 5057   × cxp 5546  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  cen 8494  Fincfn 8497   · cmul 10530  chash 13678  +gcplusg 16553  0gc0g 16701  SubGrpcsubg 18211  Cntzccntz 18383  LSSumclsm 18688  proj1cpj1 18689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-subg 18214  df-cntz 18385  df-lsm 18690  df-pj1 18691
This theorem is referenced by:  ablfacrp2  19118  ablfac1eulem  19123  ablfac1eu  19124  pgpfaclem2  19133
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