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Theorem grpinvpropd 19046
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 18688 . . . . . . . 8 (𝜑 → (0g𝐾) = (0g𝐿))
54adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
61, 5eqeq12d 2751 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
76anass1rs 655 . . . . 5 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87riotabidva 7407 . . . 4 ((𝜑𝑦𝐵) → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98mpteq2dva 5248 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))))
102riotaeqdv 7389 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
112, 10mpteq12dv 5239 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))))
123riotaeqdv 7389 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)) = (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
133, 12mpteq12dv 5239 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
149, 11, 133eqtr3d 2783 . 2 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
15 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2735 . . 3 (+g𝐾) = (+g𝐾)
17 eqid 2735 . . 3 (0g𝐾) = (0g𝐾)
18 eqid 2735 . . 3 (invg𝐾) = (invg𝐾)
1915, 16, 17, 18grpinvfval 19009 . 2 (invg𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
20 eqid 2735 . . 3 (Base‘𝐿) = (Base‘𝐿)
21 eqid 2735 . . 3 (+g𝐿) = (+g𝐿)
22 eqid 2735 . . 3 (0g𝐿) = (0g𝐿)
23 eqid 2735 . . 3 (invg𝐿) = (invg𝐿)
2420, 21, 22, 23grpinvfval 19009 . 2 (invg𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2514, 19, 243eqtr4g 2800 1 (𝜑 → (invg𝐾) = (invg𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  invgcminusg 18965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-minusg 18968
This theorem is referenced by:  grpsubpropd  19076  grpsubpropd2  19077  mulgpropd  19147  invrpropd  20435  rlmvneg  21231  matinvg  22440  tngngp3  24693
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