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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 6918 | . . . 4 ⊢ (𝐺 ∈ V → ( I ‘𝐺) = 𝐺) | |
3 | 2 | fveq2d 6847 | . . 3 ⊢ (𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
4 | base0 17093 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
5 | eqid 2733 | . . . . . 6 ⊢ (invg‘∅) = (invg‘∅) | |
6 | 4, 5 | grpinvfn 18797 | . . . . 5 ⊢ (invg‘∅) Fn ∅ |
7 | fn0 6633 | . . . . 5 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
8 | 6, 7 | mpbi 229 | . . . 4 ⊢ (invg‘∅) = ∅ |
9 | fvprc 6835 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → ( I ‘𝐺) = ∅) | |
10 | 9 | fveq2d 6847 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘∅)) |
11 | fvprc 6835 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (invg‘𝐺) = ∅) | |
12 | 8, 10, 11 | 3eqtr4a 2799 | . . 3 ⊢ (¬ 𝐺 ∈ V → (invg‘( I ‘𝐺)) = (invg‘𝐺)) |
13 | 3, 12 | pm2.61i 182 | . 2 ⊢ (invg‘( I ‘𝐺)) = (invg‘𝐺) |
14 | 1, 13 | eqtr4i 2764 | 1 ⊢ 𝑁 = (invg‘( I ‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 I cid 5531 Fn wfn 6492 ‘cfv 6497 invgcminusg 18754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-1cn 11114 ax-addcl 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-nn 12159 df-slot 17059 df-ndx 17071 df-base 17089 df-minusg 18757 |
This theorem is referenced by: deg1invg 25487 |
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