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Theorem grpinvpropd 17697
Description: If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
grpinvpropd.1 (𝜑𝐵 = (Base‘𝐾))
grpinvpropd.2 (𝜑𝐵 = (Base‘𝐿))
grpinvpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
grpinvpropd (𝜑 → (invg𝐾) = (invg𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grpinvpropd
StepHypRef Expression
1 grpinvpropd.3 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
2 grpinvpropd.1 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐾))
3 grpinvpropd.2 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝐿))
42, 3, 1grpidpropd 17468 . . . . . . . 8 (𝜑 → (0g𝐾) = (0g𝐿))
54adantr 466 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
61, 5eqeq12d 2786 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
76anass1rs 626 . . . . 5 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87riotabidva 6769 . . . 4 ((𝜑𝑦𝐵) → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98mpteq2dva 4878 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))))
102riotaeqdv 6754 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
112, 10mpteq12dv 4867 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))))
123riotaeqdv 6754 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)) = (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
133, 12mpteq12dv 4867 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
149, 11, 133eqtr3d 2813 . 2 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
15 eqid 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2771 . . 3 (+g𝐾) = (+g𝐾)
17 eqid 2771 . . 3 (0g𝐾) = (0g𝐾)
18 eqid 2771 . . 3 (invg𝐾) = (invg𝐾)
1915, 16, 17, 18grpinvfval 17667 . 2 (invg𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
20 eqid 2771 . . 3 (Base‘𝐿) = (Base‘𝐿)
21 eqid 2771 . . 3 (+g𝐿) = (+g𝐿)
22 eqid 2771 . . 3 (0g𝐿) = (0g𝐿)
23 eqid 2771 . . 3 (invg𝐿) = (invg𝐿)
2420, 21, 22, 23grpinvfval 17667 . 2 (invg𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2514, 19, 243eqtr4g 2830 1 (𝜑 → (invg𝐾) = (invg𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cmpt 4863  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  0gc0g 16307  invgcminusg 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-0g 16309  df-minusg 17633
This theorem is referenced by:  grpsubpropd  17727  grpsubpropd2  17728  mulgpropd  17791  invrpropd  18905  rlmvneg  19421  matinvg  20440  tngngp3  22679
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