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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 6733 | . . . 4 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | fveq2d 6667 | . . 3 ⊢ (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
4 | base0 16524 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2818 | . . . . . 6 ⊢ (invg‘∅) = (invg‘∅) | |
6 | 4, 5 | grpinvfn 18083 | . . . . 5 ⊢ (invg‘∅) Fn ∅ |
7 | fn0 6472 | . . . . 5 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
8 | 6, 7 | mpbi 231 | . . . 4 ⊢ (invg‘∅) = ∅ |
9 | fvprc 6656 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
10 | 9 | fveq2d 6667 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅)) |
11 | fvprc 6656 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
12 | 8, 10, 11 | 3eqtr4a 2879 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
13 | 3, 12 | pm2.61i 183 | . 2 ⊢ (invg‘( I ‘𝐺)) = (invg‘𝐺) |
14 | 1, 13 | eqtr4i 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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