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Mirrors > Home > MPE Home > Th. List > grpinvval | Structured version Visualization version GIF version |
Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.) |
Ref | Expression |
---|---|
grpinvval.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinvval.p | ⊢ + = (+g‘𝐺) |
grpinvval.o | ⊢ 0 = (0g‘𝐺) |
grpinvval.n | ⊢ 𝑁 = (invg‘𝐺) |
Ref | Expression |
---|---|
grpinvval | ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7366 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋)) | |
2 | 1 | eqeq1d 2735 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 )) |
3 | 2 | riotabidv 7316 | . 2 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
4 | grpinvval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
5 | grpinvval.p | . . 3 ⊢ + = (+g‘𝐺) | |
6 | grpinvval.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
7 | grpinvval.n | . . 3 ⊢ 𝑁 = (invg‘𝐺) | |
8 | 4, 5, 6, 7 | grpinvfval 18794 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
9 | riotaex 7318 | . 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V | |
10 | 3, 8, 9 | fvmpt 6949 | 1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6497 ℩crio 7313 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 0gc0g 17326 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 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 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-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-riota 7314 df-ov 7361 df-minusg 18757 |
This theorem is referenced by: grplinv 18805 isgrpinv 18809 xrsinvgval 31917 ringinvval 32119 ressply1invg 32328 |
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