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Theorem grpoinvval 29412
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 29411 . . 3 (𝐺 ∈ GrpOp → 𝑁 = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)))
54fveq1d 6844 . 2 (𝐺 ∈ GrpOp → (𝑁𝐴) = ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴))
6 oveq2 7364 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴))
76eqeq1d 2738 . . . 4 (𝑥 = 𝐴 → ((𝑦𝐺𝑥) = 𝑈 ↔ (𝑦𝐺𝐴) = 𝑈))
87riotabidv 7314 . . 3 (𝑥 = 𝐴 → (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
9 eqid 2736 . . 3 (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈)) = (𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))
10 riotaex 7316 . . 3 (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ V
118, 9, 10fvmpt 6948 . 2 (𝐴𝑋 → ((𝑥𝑋 ↦ (𝑦𝑋 (𝑦𝐺𝑥) = 𝑈))‘𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
125, 11sylan9eq 2796 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cmpt 5188  ran crn 5634  cfv 6496  crio 7311  (class class class)co 7356  GrpOpcgr 29378  GIdcgi 29379  invcgn 29380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7312  df-ov 7359  df-ginv 29384
This theorem is referenced by:  grpoinvcl  29413  grpoinv  29414
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