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Theorem grpinvfvi 17656
Description: The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypothesis
Ref Expression
grpinvfvi.t 𝑁 = (invg𝐺)
Assertion
Ref Expression
grpinvfvi 𝑁 = (invg‘( I ‘𝐺))

Proof of Theorem grpinvfvi
StepHypRef Expression
1 grpinvfvi.t . 2 𝑁 = (invg𝐺)
2 fvi 6409 . . . 4 (𝐺 ∈ V → ( I ‘𝐺) = 𝐺)
32fveq2d 6348 . . 3 (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
4 base0 16106 . . . . . 6 ∅ = (Base‘∅)
5 eqid 2752 . . . . . 6 (invg‘∅) = (invg‘∅)
64, 5grpinvfn 17655 . . . . 5 (invg‘∅) Fn ∅
7 fn0 6164 . . . . 5 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
86, 7mpbi 220 . . . 4 (invg‘∅) = ∅
9 fvprc 6338 . . . . 5 𝐺 ∈ V → ( I ‘𝐺) = ∅)
109fveq2d 6348 . . . 4 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅))
11 fvprc 6338 . . . 4 𝐺 ∈ V → (invg𝐺) = ∅)
128, 10, 113eqtr4a 2812 . . 3 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
133, 12pm2.61i 176 . 2 (invg‘( I ‘𝐺)) = (invg𝐺)
141, 13eqtr4i 2777 1 𝑁 = (invg‘( I ‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  c0 4050   I cid 5165   Fn wfn 6036  cfv 6041  invgcminusg 17616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-slot 16055  df-base 16057  df-minusg 17619
This theorem is referenced by:  deg1invg  24057
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