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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 479 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
73, 6eqeq12d 2746 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87anass1rs 651 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98rexbidva 3174 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
109ralbidva 3173 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
111rexeqdv 3324 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
121, 11raleqbidv 3340 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
132rexeqdv 3324 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
142, 13raleqbidv 3340 . . . 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 629 . 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 2730 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2730 . . 3 (+g𝐾) = (+g𝐾)
19 eqid 2730 . . 3 (0g𝐾) = (0g𝐾)
2017, 18, 19isgrp 18861 . 2 (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
21 eqid 2730 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2730 . . 3 (+g𝐿) = (+g𝐿)
23 eqid 2730 . . 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 394   = wceq 1539  wcel 2104  wral 3059  wrex 3068  cfv 6542  (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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  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  33245
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