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Theorem grpinvfvi 17384
 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 6212 . . . 4 (𝐺 ∈ V → ( I ‘𝐺) = 𝐺)
32fveq2d 6152 . . 3 (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
4 base0 15833 . . . . . 6 ∅ = (Base‘∅)
5 eqid 2621 . . . . . 6 (invg‘∅) = (invg‘∅)
64, 5grpinvfn 17383 . . . . 5 (invg‘∅) Fn ∅
7 fn0 5968 . . . . 5 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
86, 7mpbi 220 . . . 4 (invg‘∅) = ∅
9 fvprc 6142 . . . . 5 𝐺 ∈ V → ( I ‘𝐺) = ∅)
109fveq2d 6152 . . . 4 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅))
11 fvprc 6142 . . . 4 𝐺 ∈ V → (invg𝐺) = ∅)
128, 10, 113eqtr4a 2681 . . 3 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
133, 12pm2.61i 176 . 2 (invg‘( I ‘𝐺)) = (invg𝐺)
141, 13eqtr4i 2646 1 𝑁 = (invg‘( I ‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∅c0 3891   I cid 4984   Fn wfn 5842  ‘cfv 5847  invgcminusg 17344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-slot 15785  df-base 15786  df-minusg 17347 This theorem is referenced by:  deg1invg  23770
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