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Theorem grpidpropd 17182
Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
Hypotheses
Ref Expression
grpidpropd.1 (𝜑𝐵 = (Base‘𝐾))
grpidpropd.2 (𝜑𝐵 = (Base‘𝐿))
grpidpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
grpidpropd (𝜑 → (0g𝐾) = (0g𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem grpidpropd
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpidpropd.3 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
21eqeq1d 2623 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = 𝑦 ↔ (𝑥(+g𝐿)𝑦) = 𝑦))
31oveqrspc2v 6627 . . . . . . . . . . 11 ((𝜑 ∧ (𝑧𝐵𝑤𝐵)) → (𝑧(+g𝐾)𝑤) = (𝑧(+g𝐿)𝑤))
43oveqrspc2v 6627 . . . . . . . . . 10 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
54ancom2s 843 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐾)𝑥) = (𝑦(+g𝐿)𝑥))
65eqeq1d 2623 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(+g𝐾)𝑥) = 𝑦 ↔ (𝑦(+g𝐿)𝑥) = 𝑦))
72, 6anbi12d 746 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
87anassrs 679 . . . . . 6 (((𝜑𝑥𝐵) ∧ 𝑦𝐵) → (((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
98ralbidva 2979 . . . . 5 ((𝜑𝑥𝐵) → (∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
109pm5.32da 672 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
11 grpidpropd.1 . . . . . 6 (𝜑𝐵 = (Base‘𝐾))
1211eleq2d 2684 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐾)))
1311raleqdv 3133 . . . . 5 (𝜑 → (∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)))
1412, 13anbi12d 746 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))))
15 grpidpropd.2 . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
1615eleq2d 2684 . . . . 5 (𝜑 → (𝑥𝐵𝑥 ∈ (Base‘𝐿)))
1715raleqdv 3133 . . . . 5 (𝜑 → (∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
1816, 17anbi12d 746 . . . 4 (𝜑 → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
1910, 14, 183bitr3d 298 . . 3 (𝜑 → ((𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
2019iotabidv 5831 . 2 (𝜑 → (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦))))
21 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
22 eqid 2621 . . 3 (+g𝐾) = (+g𝐾)
23 eqid 2621 . . 3 (0g𝐾) = (0g𝐾)
2421, 22, 23grpidval 17181 . 2 (0g𝐾) = (℩𝑥(𝑥 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ (Base‘𝐾)((𝑥(+g𝐾)𝑦) = 𝑦 ∧ (𝑦(+g𝐾)𝑥) = 𝑦)))
25 eqid 2621 . . 3 (Base‘𝐿) = (Base‘𝐿)
26 eqid 2621 . . 3 (+g𝐿) = (+g𝐿)
27 eqid 2621 . . 3 (0g𝐿) = (0g𝐿)
2825, 26, 27grpidval 17181 . 2 (0g𝐿) = (℩𝑥(𝑥 ∈ (Base‘𝐿) ∧ ∀𝑦 ∈ (Base‘𝐿)((𝑥(+g𝐿)𝑦) = 𝑦 ∧ (𝑦(+g𝐿)𝑥) = 𝑦)))
2920, 24, 283eqtr4g 2680 1 (𝜑 → (0g𝐾) = (0g𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cio 5808  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-0g 16023
This theorem is referenced by:  gsumpropd  17193  gsumpropd2lem  17194  mhmpropd  17262  grppropd  17358  grpinvpropd  17411  mulgpropd  17505  prds1  18535  rngidpropd  18616  drngprop  18679  drngpropd  18695  abvpropd  18763  lbspropd  19018  sralmod0  19107  opsr0  19507  mplbaspropd  19526  ply1mpl0  19544  phlpropd  19919  mat0  20142  nmpropd  22308  nmpropd2  22309  tng0  22357  mdegpropd  23748  ply1divalg2  23802  resv0g  29618  zlm0  29785  hlhils0  36714  hlhil0  36724
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