MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grppropd Structured version   Visualization version   GIF version

Theorem grppropd 17377
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 17256 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
51, 2, 3grpidpropd 17201 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
65adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
73, 6eqeq12d 2636 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87anass1rs 848 . . . . . 6 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98rexbidva 3044 . . . . 5 ((𝜑𝑦𝐵) → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
109ralbidva 2981 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
111rexeqdv 3138 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
121, 11raleqbidv 3145 . . . 4 (𝜑 → (∀𝑦𝐵𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
132rexeqdv 3138 . . . . 5 (𝜑 → (∃𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
142, 13raleqbidv 3145 . . . 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 746 . 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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
18 eqid 2621 . . 3 (+g𝐾) = (+g𝐾)
19 eqid 2621 . . 3 (0g𝐾) = (0g𝐾)
2017, 18, 19isgrp 17368 . 2 (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
21 eqid 2621 . . 3 (Base‘𝐿) = (Base‘𝐿)
22 eqid 2621 . . 3 (+g𝐿) = (+g𝐿)
23 eqid 2621 . . 3 (0g𝐿) = (0g𝐿)
2421, 22, 23isgrp 17368 . 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 384   = wceq 1480  wcel 1987  wral 2908  wrex 2909  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  0gc0g 16040  Mndcmnd 17234  Grpcgrp 17362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365
This theorem is referenced by:  grpprop  17378  ghmpropd  17638  oppggrpb  17728  ablpropd  18143  ringpropd  18522  lmodprop2d  18865  sralmod  19127  nmpropd2  22339  ngppropd  22381  tngngp2  22396  tnggrpr  22399  zhmnrg  29835
  Copyright terms: Public domain W3C validator