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Theorem grppropd 17559
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 17438 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
51, 2, 3grpidpropd 17383 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
65adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
73, 6eqeq12d 2739 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87anass1rs 884 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98rexbidva 3151 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
109ralbidva 3087 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
111rexeqdv 3248 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
121, 11raleqbidv 3255 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
132rexeqdv 3248 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
142, 13raleqbidv 3255 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
1510, 12, 143bitr3d 298 . . 3 (𝜑 → (∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
164, 15anbi12d 749 . 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 2724 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2724 . . 3 (+g𝐾) = (+g𝐾)
19 eqid 2724 . . 3 (0g𝐾) = (0g𝐾)
2017, 18, 19isgrp 17550 . 2 (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
21 eqid 2724 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2724 . . 3 (+g𝐿) = (+g𝐿)
23 eqid 2724 . . 3 (0g𝐿) = (0g𝐿)
2421, 22, 23isgrp 17550 . 2 (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2516, 20, 243bitr4g 303 1 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  cfv 6001  (class class class)co 6765  Basecbs 15980  +gcplusg 16064  0gc0g 16223  Mndcmnd 17416  Grpcgrp 17544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-0g 16225  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-grp 17547
This theorem is referenced by:  grpprop  17560  ghmpropd  17820  oppggrpb  17909  ablpropd  18324  ringpropd  18703  lmodprop2d  19048  sralmod  19310  nmpropd2  22521  ngppropd  22563  tngngp2  22578  tnggrpr  22581  zhmnrg  30241
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