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Theorem grpinvfvi 17923
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 6562 . . . 4 (𝐺 ∈ V → ( I ‘𝐺) = 𝐺)
32fveq2d 6497 . . 3 (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
4 base0 16382 . . . . . 6 ∅ = (Base‘∅)
5 eqid 2772 . . . . . 6 (invg‘∅) = (invg‘∅)
64, 5grpinvfn 17922 . . . . 5 (invg‘∅) Fn ∅
7 fn0 6303 . . . . 5 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
86, 7mpbi 222 . . . 4 (invg‘∅) = ∅
9 fvprc 6486 . . . . 5 𝐺 ∈ V → ( I ‘𝐺) = ∅)
109fveq2d 6497 . . . 4 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅))
11 fvprc 6486 . . . 4 𝐺 ∈ V → (invg𝐺) = ∅)
128, 10, 113eqtr4a 2834 . . 3 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
133, 12pm2.61i 177 . 2 (invg‘( I ‘𝐺)) = (invg𝐺)
141, 13eqtr4i 2799 1 𝑁 = (invg‘( I ‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2048  Vcvv 3409  c0 4173   I cid 5304   Fn wfn 6177  cfv 6182  invgcminusg 17882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-riota 6931  df-ov 6973  df-slot 16333  df-base 16335  df-minusg 17885
This theorem is referenced by:  deg1invg  24393
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