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Theorem lsmhash 19487
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 7392 . . 3 (𝜑 → (𝑇 𝑈) ∈ V)
2 eqid 2736 . . . 4 (𝑥 ∈ (𝑇 𝑈) ↦ ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩) = (𝑥 ∈ (𝑇 𝑈) ↦ ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩)
3 eqid 2736 . . . . . . 7 (+g𝐺) = (+g𝐺)
4 lsmhash.p . . . . . . 7 = (LSSum‘𝐺)
5 lsmhash.o . . . . . . 7 0 = (0g𝐺)
6 lsmhash.z . . . . . . 7 𝑍 = (Cntz‘𝐺)
7 lsmhash.t . . . . . . 7 (𝜑𝑇 ∈ (SubGrp‘𝐺))
8 lsmhash.u . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
9 lsmhash.i . . . . . . 7 (𝜑 → (𝑇𝑈) = { 0 })
10 lsmhash.s . . . . . . 7 (𝜑𝑇 ⊆ (𝑍𝑈))
11 eqid 2736 . . . . . . 7 (proj1𝐺) = (proj1𝐺)
123, 4, 5, 6, 7, 8, 9, 10, 11pj1f 19479 . . . . . 6 (𝜑 → (𝑇(proj1𝐺)𝑈):(𝑇 𝑈)⟶𝑇)
1312ffvelcdmda 7035 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇)
143, 4, 5, 6, 7, 8, 9, 10, 11pj2f 19480 . . . . . 6 (𝜑 → (𝑈(proj1𝐺)𝑇):(𝑇 𝑈)⟶𝑈)
1514ffvelcdmda 7035 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈)
1613, 15opelxpd 5671 . . . 4 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
177, 8jca 512 . . . . 5 (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
18 xp1st 7953 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → (1st𝑦) ∈ 𝑇)
19 xp2nd 7954 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → (2nd𝑦) ∈ 𝑈)
2018, 19jca 512 . . . . 5 (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈))
213, 4lsmelvali 19432 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2217, 20, 21syl2an 596 . . . 4 ((𝜑𝑦 ∈ (𝑇 × 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
237adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
248adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
259adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇𝑈) = { 0 })
2610adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
27 simprl 769 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 𝑈))
2818ad2antll 727 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st𝑦) ∈ 𝑇)
2919ad2antll 727 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd𝑦) ∈ 𝑈)
303, 4, 5, 6, 23, 24, 25, 26, 11, 27, 28, 29pj1eq 19482 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦))))
31 eqcom 2743 . . . . . . 7 (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ↔ (1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥))
32 eqcom 2743 . . . . . . 7 (((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦) ↔ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))
3331, 32anbi12i 627 . . . . . 6 ((((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥)))
3430, 33bitrdi 286 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
35 eqop 7963 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
3635ad2antll 727 . . . . 5 ((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))))
3734, 36bitr4d 281 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩))
382, 16, 22, 37f1o2d 7607 . . 3 (𝜑 → (𝑥 ∈ (𝑇 𝑈) ↦ ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩):(𝑇 𝑈)–1-1-onto→(𝑇 × 𝑈))
391, 38hasheqf1od 14253 . 2 (𝜑 → (♯‘(𝑇 𝑈)) = (♯‘(𝑇 × 𝑈)))
40 lsmhash.1 . . 3 (𝜑𝑇 ∈ Fin)
41 lsmhash.2 . . 3 (𝜑𝑈 ∈ Fin)
42 hashxp 14334 . . 3 ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4340, 41, 42syl2anc 584 . 2 (𝜑 → (♯‘(𝑇 × 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
4439, 43eqtrd 2776 1 (𝜑 → (♯‘(𝑇 𝑈)) = ((♯‘𝑇) · (♯‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cin 3909  wss 3910  {csn 4586  cop 4592  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883   · cmul 11056  chash 14230  +gcplusg 17133  0gc0g 17321  SubGrpcsubg 18922  Cntzccntz 19095  LSSumclsm 19416  proj1cpj1 19417
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-pj1 19419
This theorem is referenced by:  ablfacrp2  19846  ablfac1eulem  19851  ablfac1eu  19852  pgpfaclem2  19861
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