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Theorem lsmhash 18502
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 6956 . . . 4 (𝜑 → (𝑇 𝑈) ∈ V)
2 lsmhash.t . . . . 5 (𝜑𝑇 ∈ (SubGrp‘𝐺))
3 lsmhash.u . . . . 5 (𝜑𝑈 ∈ (SubGrp‘𝐺))
42, 3xpexd 7238 . . . 4 (𝜑 → (𝑇 × 𝑈) ∈ V)
5 eqid 2778 . . . . . . . 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 2778 . . . . . . . 8 (proj1𝐺) = (proj1𝐺)
125, 6, 7, 8, 2, 3, 9, 10, 11pj1f 18494 . . . . . . 7 (𝜑 → (𝑇(proj1𝐺)𝑈):(𝑇 𝑈)⟶𝑇)
1312ffvelrnda 6623 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇)
145, 6, 7, 8, 2, 3, 9, 10, 11pj2f 18495 . . . . . . 7 (𝜑 → (𝑈(proj1𝐺)𝑇):(𝑇 𝑈)⟶𝑈)
1514ffvelrnda 6623 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈)
16 opelxpi 5392 . . . . . 6 ((((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
1713, 15, 16syl2anc 579 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
1817ex 403 . . . 4 (𝜑 → (𝑥 ∈ (𝑇 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈)))
192, 3jca 507 . . . . . 6 (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
20 xp1st 7477 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (1st𝑦) ∈ 𝑇)
21 xp2nd 7478 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (2nd𝑦) ∈ 𝑈)
2220, 21jca 507 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈))
235, 6lsmelvali 18449 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2419, 22, 23syl2an 589 . . . . 5 ((𝜑𝑦 ∈ (𝑇 × 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2524ex 403 . . . 4 (𝜑 → (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈)))
262adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
273adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
289adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇𝑈) = { 0 })
2910adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
30 simprl 761 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 𝑈))
3120ad2antll 719 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st𝑦) ∈ 𝑇)
3221ad2antll 719 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd𝑦) ∈ 𝑈)
335, 6, 7, 8, 26, 27, 28, 29, 11, 30, 31, 32pj1eq 18497 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦))))
34 eqcom 2785 . . . . . . . 8 (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ↔ (1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥))
35 eqcom 2785 . . . . . . . 8 (((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦) ↔ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))
3634, 35anbi12i 620 . . . . . . 7 ((((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥)))
3733, 36syl6bb 279 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
38 eqop 7487 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
3938ad2antll 719 . . . . . 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))))
4037, 39bitr4d 274 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩))
4140ex 403 . . . 4 (𝜑 → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈)) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩)))
421, 4, 18, 25, 41en3d 8278 . . 3 (𝜑 → (𝑇 𝑈) ≈ (𝑇 × 𝑈))
43 hasheni 13453 . . 3 ((𝑇 𝑈) ≈ (𝑇 × 𝑈) → (♯‘(𝑇 𝑈)) = (♯‘(𝑇 × 𝑈)))
4442, 43syl 17 . 2 (𝜑 → (♯‘(𝑇 𝑈)) = (♯‘(𝑇 × 𝑈)))
45 lsmhash.1 . . 3 (𝜑𝑇 ∈ Fin)
46 lsmhash.2 . . 3 (𝜑𝑈 ∈ Fin)
47 hashxp 13535 . . 3 ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4845, 46, 47syl2anc 579 . 2 (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4944, 48eqtrd 2814 1 (𝜑 → (♯‘(𝑇 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  cin 3791  wss 3792  {csn 4398  cop 4404   class class class wbr 4886   × cxp 5353  cfv 6135  (class class class)co 6922  1st c1st 7443  2nd c2nd 7444  cen 8238  Fincfn 8241   · cmul 10277  chash 13435  +gcplusg 16338  0gc0g 16486  SubGrpcsubg 17972  Cntzccntz 18131  LSSumclsm 18433  proj1cpj1 18434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813  df-sbg 17814  df-subg 17975  df-cntz 18133  df-lsm 18435  df-pj1 18436
This theorem is referenced by:  ablfacrp2  18853  ablfac1eulem  18858  ablfac1eu  18859  pgpfaclem2  18868
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