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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 6969 | . . . 4 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | fveq2d 6896 | . . 3 ⊢ (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
4 | base0 17184 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2725 | . . . . . 6 ⊢ (invg‘∅) = (invg‘∅) | |
6 | 4, 5 | grpinvfn 18942 | . . . . 5 ⊢ (invg‘∅) Fn ∅ |
7 | fn0 6681 | . . . . 5 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
8 | 6, 7 | mpbi 229 | . . . 4 ⊢ (invg‘∅) = ∅ |
9 | fvprc 6884 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
10 | 9 | fveq2d 6896 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅)) |
11 | fvprc 6884 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
12 | 8, 10, 11 | 3eqtr4a 2791 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
13 | 3, 12 | pm2.61i 182 | . 2 ⊢ (invg‘( I ‘𝐺)) = (invg‘𝐺) |
14 | 1, 13 | eqtr4i 2756 | 1 ⊢ 𝑁 = (invg‘( I ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∅c0 4318 I cid 5569 Fn wfn 6538 ‘cfv 6543 invgcminusg 18895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-1cn 11196 ax-addcl 11198 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-nn 12243 df-slot 17150 df-ndx 17162 df-base 17180 df-minusg 18898 |
This theorem is referenced by: deg1invg 26060 |
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