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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 = (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 6577 . . . . 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 6858 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑇 × 𝑈) ∈ V)
63, 4, 5syl2anc 691 . . . 4 (𝜑 → (𝑇 × 𝑈) ∈ V)
7 eqid 2610 . . . . . . . 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 2610 . . . . . . . 8 (proj1𝐺) = (proj1𝐺)
147, 8, 9, 10, 3, 4, 11, 12, 13pj1f 17933 . . . . . . 7 (𝜑 → (𝑇(proj1𝐺)𝑈):(𝑇 𝑈)⟶𝑇)
1514ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇)
167, 8, 9, 10, 3, 4, 11, 12, 13pj2f 17934 . . . . . . 7 (𝜑 → (𝑈(proj1𝐺)𝑇):(𝑇 𝑈)⟶𝑈)
1716ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈)
18 opelxpi 5072 . . . . . 6 ((((𝑇(proj1𝐺)𝑈)‘𝑥) ∈ 𝑇 ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) ∈ 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
1915, 17, 18syl2anc 691 . . . . 5 ((𝜑𝑥 ∈ (𝑇 𝑈)) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈))
2019ex 449 . . . 4 (𝜑 → (𝑥 ∈ (𝑇 𝑈) → ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ∈ (𝑇 × 𝑈)))
213, 4jca 553 . . . . . 6 (𝜑 → (𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)))
22 xp1st 7089 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (1st𝑦) ∈ 𝑇)
23 xp2nd 7090 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (2nd𝑦) ∈ 𝑈)
2422, 23jca 553 . . . . . 6 (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈))
257, 8lsmelvali 17888 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ ((1st𝑦) ∈ 𝑇 ∧ (2nd𝑦) ∈ 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2621, 24, 25syl2an 493 . . . . 5 ((𝜑𝑦 ∈ (𝑇 × 𝑈)) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈))
2726ex 449 . . . 4 (𝜑 → (𝑦 ∈ (𝑇 × 𝑈) → ((1st𝑦)(+g𝐺)(2nd𝑦)) ∈ (𝑇 𝑈)))
283adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ∈ (SubGrp‘𝐺))
294adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑈 ∈ (SubGrp‘𝐺))
3011adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑇𝑈) = { 0 })
3112adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑇 ⊆ (𝑍𝑈))
32 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → 𝑥 ∈ (𝑇 𝑈))
3322ad2antll 761 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (1st𝑦) ∈ 𝑇)
3423ad2antll 761 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (2nd𝑦) ∈ 𝑈)
357, 8, 9, 10, 28, 29, 30, 31, 13, 32, 33, 34pj1eq 17936 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦))))
36 eqcom 2617 . . . . . . . 8 (((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ↔ (1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥))
37 eqcom 2617 . . . . . . . 8 (((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦) ↔ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))
3836, 37anbi12i 729 . . . . . . 7 ((((𝑇(proj1𝐺)𝑈)‘𝑥) = (1st𝑦) ∧ ((𝑈(proj1𝐺)𝑇)‘𝑥) = (2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥)))
3935, 38syl6bb 275 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
40 eqop 7099 . . . . . . 7 (𝑦 ∈ (𝑇 × 𝑈) → (𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩ ↔ ((1st𝑦) = ((𝑇(proj1𝐺)𝑈)‘𝑥) ∧ (2nd𝑦) = ((𝑈(proj1𝐺)𝑇)‘𝑥))))
4140ad2antll 761 . . . . . 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 270 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 × 𝑈))) → (𝑥 = ((1st𝑦)(+g𝐺)(2nd𝑦)) ↔ 𝑦 = ⟨((𝑇(proj1𝐺)𝑈)‘𝑥), ((𝑈(proj1𝐺)𝑇)‘𝑥)⟩))
4342ex 449 . . . 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 7878 . . 3 (𝜑 → (𝑇 𝑈) ≈ (𝑇 × 𝑈))
45 hasheni 12998 . . 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 13081 . . 3 ((𝑇 ∈ Fin ∧ 𝑈 ∈ Fin) → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
5047, 48, 49syl2anc 691 . 2 (𝜑 → (#‘(𝑇 × 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
5146, 50eqtrd 2644 1 (𝜑 → (#‘(𝑇 𝑈)) = ((#‘𝑇) · (#‘𝑈)))
 Colors of variables: wff setvar class 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  1st c1st 7057  2nd c2nd 7058   ≈ cen 7838  Fincfn 7841   · cmul 9820  #chash 12979  +gcplusg 15768  0gc0g 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
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