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Theorem grpinvfvi 18084
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 6733 . . . 4 (𝐺 ∈ V → ( I ‘𝐺) = 𝐺)
32fveq2d 6667 . . 3 (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
4 base0 16524 . . . . . 6 ∅ = (Base‘∅)
5 eqid 2818 . . . . . 6 (invg‘∅) = (invg‘∅)
64, 5grpinvfn 18083 . . . . 5 (invg‘∅) Fn ∅
7 fn0 6472 . . . . 5 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
86, 7mpbi 231 . . . 4 (invg‘∅) = ∅
9 fvprc 6656 . . . . 5 𝐺 ∈ V → ( I ‘𝐺) = ∅)
109fveq2d 6667 . . . 4 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅))
11 fvprc 6656 . . . 4 𝐺 ∈ V → (invg𝐺) = ∅)
128, 10, 113eqtr4a 2879 . . 3 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
133, 12pm2.61i 183 . 2 (invg‘( I ‘𝐺)) = (invg𝐺)
141, 13eqtr4i 2844 1 𝑁 = (invg‘( I ‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288   I cid 5452   Fn wfn 6343  cfv 6348  invgcminusg 18042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7103  df-ov 7148  df-slot 16475  df-base 16477  df-minusg 18045
This theorem is referenced by:  deg1invg  24627
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