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Theorem grppropd 18873
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 18684 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
51, 2, 3grpidpropd 18587 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
65adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
73, 6eqeq12d 2748 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87anass1rs 653 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98rexbidva 3176 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
109ralbidva 3175 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
111rexeqdv 3326 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
121, 11raleqbidv 3342 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
132rexeqdv 3326 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
142, 13raleqbidv 3342 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
1510, 12, 143bitr3d 308 . . 3 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
164, 15anbi12d 631 . 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 2732 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2732 . . 3 (+g𝐾) = (+g𝐾)
19 eqid 2732 . . 3 (0g𝐾) = (0g𝐾)
2017, 18, 19isgrp 18861 . 2 (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
21 eqid 2732 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2732 . . 3 (+g𝐿) = (+g𝐿)
23 eqid 2732 . . 3 (0g𝐿) = (0g𝐿)
2421, 22, 23isgrp 18861 . 2 (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2516, 20, 243bitr4g 313 1 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cfv 6543  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  0gc0g 17389  Mndcmnd 18659  Grpcgrp 18855
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858
This theorem is referenced by:  grpprop  18874  ghmpropd  19170  oppggrpb  19266  ablpropd  19701  ringpropd  20176  lmodprop2d  20678  sralmod  20954  psrgrp  21736  nmpropd2  24324  ngppropd  24366  tngngp2  24389  tnggrpr  24392  zhmnrg  33233
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