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Mirrors > Home > MPE Home > Th. List > grpinvfvi | Structured version Visualization version GIF version |
Description: The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
grpinvfvi.t | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvfvi | ⊢ 𝑁 = (invg‘( I ‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvfvi.t | . 2 ⊢ 𝑁 = (invg‘𝐺) | |
2 | fvi 6562 | . . . 4 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | fveq2d 6497 | . . 3 ⊢ (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
4 | base0 16382 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2772 | . . . . . 6 ⊢ (invg‘∅) = (invg‘∅) | |
6 | 4, 5 | grpinvfn 17922 | . . . . 5 ⊢ (invg‘∅) Fn ∅ |
7 | fn0 6303 | . . . . 5 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
8 | 6, 7 | mpbi 222 | . . . 4 ⊢ (invg‘∅) = ∅ |
9 | fvprc 6486 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
10 | 9 | fveq2d 6497 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅)) |
11 | fvprc 6486 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
12 | 8, 10, 11 | 3eqtr4a 2834 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
13 | 3, 12 | pm2.61i 177 | . 2 ⊢ (invg‘( I ‘𝐺)) = (invg‘𝐺) |
14 | 1, 13 | eqtr4i 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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