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Theorem grpinvpropd 17806
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 17576 . . . . . . . 8 (𝜑 → (0g𝐾) = (0g𝐿))
54adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (0g𝐾) = (0g𝐿))
61, 5eqeq12d 2814 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
76anass1rs 646 . . . . 5 (((𝜑𝑦𝐵) ∧ 𝑥𝐵) → ((𝑥(+g𝐾)𝑦) = (0g𝐾) ↔ (𝑥(+g𝐿)𝑦) = (0g𝐿)))
87riotabidva 6855 . . . 4 ((𝜑𝑦𝐵) → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)))
98mpteq2dva 4937 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))))
102riotaeqdv 6840 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾)) = (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
112, 10mpteq12dv 4926 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))))
123riotaeqdv 6840 . . . 4 (𝜑 → (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿)) = (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
133, 12mpteq12dv 4926 . . 3 (𝜑 → (𝑦𝐵 ↦ (𝑥𝐵 (𝑥(+g𝐿)𝑦) = (0g𝐿))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
149, 11, 133eqtr3d 2841 . 2 (𝜑 → (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾))) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿))))
15 eqid 2799 . . 3 (Base‘𝐾) = (Base‘𝐾)
16 eqid 2799 . . 3 (+g𝐾) = (+g𝐾)
17 eqid 2799 . . 3 (0g𝐾) = (0g𝐾)
18 eqid 2799 . . 3 (invg𝐾) = (invg𝐾)
1915, 16, 17, 18grpinvfval 17776 . 2 (invg𝐾) = (𝑦 ∈ (Base‘𝐾) ↦ (𝑥 ∈ (Base‘𝐾)(𝑥(+g𝐾)𝑦) = (0g𝐾)))
20 eqid 2799 . . 3 (Base‘𝐿) = (Base‘𝐿)
21 eqid 2799 . . 3 (+g𝐿) = (+g𝐿)
22 eqid 2799 . . 3 (0g𝐿) = (0g𝐿)
23 eqid 2799 . . 3 (invg𝐿) = (invg𝐿)
2420, 21, 22, 23grpinvfval 17776 . 2 (invg𝐿) = (𝑦 ∈ (Base‘𝐿) ↦ (𝑥 ∈ (Base‘𝐿)(𝑥(+g𝐿)𝑦) = (0g𝐿)))
2514, 19, 243eqtr4g 2858 1 (𝜑 → (invg𝐾) = (invg𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cmpt 4922  cfv 6101  crio 6838  (class class class)co 6878  Basecbs 16184  +gcplusg 16267  0gc0g 16415  invgcminusg 17739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-0g 16417  df-minusg 17742
This theorem is referenced by:  grpsubpropd  17836  grpsubpropd2  17837  mulgpropd  17897  invrpropd  19014  rlmvneg  19530  matinvg  20549  tngngp3  22788
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