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Mirrors > Home > MPE Home > Th. List > grpoinvcl | Structured version Visualization version GIF version |
Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpinvcl.1 | ⊢ 𝑋 = ran 𝐺 |
grpinvcl.2 | ⊢ 𝑁 = (inv‘𝐺) |
Ref | Expression |
---|---|
grpoinvcl | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinvcl.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | eqid 2798 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
3 | grpinvcl.2 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinvval 28306 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))) |
5 | 1, 2 | grpoinveu 28302 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) |
6 | riotacl 7110 | . . 3 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋) | |
7 | 5, 6 | syl 17 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋) |
8 | 4, 7 | eqeltrd 2890 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃!wreu 3108 ran crn 5520 ‘cfv 6324 ℩crio 7092 (class class class)co 7135 GrpOpcgr 28272 GIdcgi 28273 invcgn 28274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-grpo 28276 df-gid 28277 df-ginv 28278 |
This theorem is referenced by: grpoinvid1 28311 grpoinvid2 28312 grpolcan 28313 grpo2inv 28314 grpoinvf 28315 grpoinvop 28316 grpodivinv 28319 grpoinvdiv 28320 grpodivf 28321 grpomuldivass 28324 grponpcan 28326 ablodivdiv4 28337 vcm 28359 rngonegcl 35365 isdrngo2 35396 |
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