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Theorem grpinvfvi 18141
 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 6715 . . . 4 (𝐺 ∈ V → ( I ‘𝐺) = 𝐺)
32fveq2d 6649 . . 3 (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
4 base0 16530 . . . . . 6 ∅ = (Base‘∅)
5 eqid 2798 . . . . . 6 (invg‘∅) = (invg‘∅)
64, 5grpinvfn 18140 . . . . 5 (invg‘∅) Fn ∅
7 fn0 6451 . . . . 5 ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
86, 7mpbi 233 . . . 4 (invg‘∅) = ∅
9 fvprc 6638 . . . . 5 𝐺 ∈ V → ( I ‘𝐺) = ∅)
109fveq2d 6649 . . . 4 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅))
11 fvprc 6638 . . . 4 𝐺 ∈ V → (invg𝐺) = ∅)
128, 10, 113eqtr4a 2859 . . 3 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg𝐺))
133, 12pm2.61i 185 . 2 (invg‘( I ‘𝐺)) = (invg𝐺)
141, 13eqtr4i 2824 1 𝑁 = (invg‘( I ‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243   I cid 5424   Fn wfn 6319  ‘cfv 6324  invgcminusg 18098 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-riota 7093  df-ov 7138  df-slot 16481  df-base 16483  df-minusg 18101 This theorem is referenced by:  deg1invg  24714
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