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Theorem grpoinvval 30451
Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvfval.1 𝑋 = ran 𝐺
grpinvfval.2 𝑈 = (GId‘𝐺)
grpinvfval.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvval ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑋
Allowed substitution hints:   𝑈(𝑦)   𝑁(𝑦)

Proof of Theorem grpoinvval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 grpinvfval.1 . . . 4 𝑋 = ran 𝐺
2 grpinvfval.2 . . . 4 𝑈 = (GId‘𝐺)
3 grpinvfval.3 . . . 4 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvfval 30450 . . 3 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
54fveq1d 6893 . 2 (𝐺 ∈ GrpOp → (𝑁𝐴) = ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6 oveq2 7422 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
76eqeq1d 2728 . . . 4 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
87riotabidv 7372 . . 3 (𝑥 = 𝐴 → (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
9 eqid 2726 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
10 riotaex 7374 . . 3 (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
118, 9, 10fvmpt 6999 . 2 (𝐴𝑋 → ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
125, 11sylan9eq 2786 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cmpt 5227  ran crn 5674  cfv 6544  crio 7369  (class class class)co 7414  GrpOpcgr 30417  GIdcgi 30418  invcgn 30419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-ginv 30423
This theorem is referenced by:  grpoinvcl  30452  grpoinv  30453
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