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Theorem grppropd 17824
Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
grppropd.1 (𝜑𝐵 = (Base‘𝐾))
grppropd.2 (𝜑𝐵 = (Base‘𝐿))
grppropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
grppropd (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grppropd
StepHypRef Expression
1 grppropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 grppropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 grppropd.3 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
41, 2, 3mndpropd 17702 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
51, 2, 3grpidpropd 17647 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
65adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
73, 6eqeq12d 2792 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87anass1rs 645 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98rexbidva 3233 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
109ralbidva 3166 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
111rexeqdv 3340 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
121, 11raleqbidv 3325 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
132rexeqdv 3340 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
142, 13raleqbidv 3325 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
1510, 12, 143bitr3d 301 . . 3 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
164, 15anbi12d 624 . 2 (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
17 eqid 2777 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2777 . . 3 (+g𝐾) = (+g𝐾)
19 eqid 2777 . . 3 (0g𝐾) = (0g𝐾)
2017, 18, 19isgrp 17815 . 2 (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
21 eqid 2777 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2777 . . 3 (+g𝐿) = (+g𝐿)
23 eqid 2777 . . 3 (0g𝐿) = (0g𝐿)
2421, 22, 23isgrp 17815 . 2 (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2516, 20, 243bitr4g 306 1 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2106  wral 3089  wrex 3090  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  0gc0g 16486  Mndcmnd 17680  Grpcgrp 17809
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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812
This theorem is referenced by:  grpprop  17825  ghmpropd  18082  oppggrpb  18171  ablpropd  18589  ringpropd  18969  lmodprop2d  19317  sralmod  19584  nmpropd2  22807  ngppropd  22849  tngngp2  22864  tnggrpr  22867  zhmnrg  30609
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