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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 6740 | . . . 4 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | fveq2d 6674 | . . 3 ⊢ (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
4 | base0 16536 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2821 | . . . . . 6 ⊢ (invg‘∅) = (invg‘∅) | |
6 | 4, 5 | grpinvfn 18145 | . . . . 5 ⊢ (invg‘∅) Fn ∅ |
7 | fn0 6479 | . . . . 5 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
8 | 6, 7 | mpbi 232 | . . . 4 ⊢ (invg‘∅) = ∅ |
9 | fvprc 6663 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
10 | 9 | fveq2d 6674 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅)) |
11 | fvprc 6663 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
12 | 8, 10, 11 | 3eqtr4a 2882 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
13 | 3, 12 | pm2.61i 184 | . 2 ⊢ (invg‘( I ‘𝐺)) = (invg‘𝐺) |
14 | 1, 13 | eqtr4i 2847 | 1 ⊢ 𝑁 = (invg‘( I ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 I cid 5459 Fn wfn 6350 ‘cfv 6355 invgcminusg 18104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-slot 16487 df-base 16489 df-minusg 18107 |
This theorem is referenced by: deg1invg 24700 |
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