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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 2823 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
3 | grpinvcl.2 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinvval 28302 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))) |
5 | 1, 2 | grpoinveu 28298 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) |
6 | riotacl 7133 | . . 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 2915 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃!wreu 3142 ran crn 5558 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 GrpOpcgr 28268 GIdcgi 28269 invcgn 28270 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-grpo 28272 df-gid 28273 df-ginv 28274 |
This theorem is referenced by: grpoinvid1 28307 grpoinvid2 28308 grpolcan 28309 grpo2inv 28310 grpoinvf 28311 grpoinvop 28312 grpodivinv 28315 grpoinvdiv 28316 grpodivf 28317 grpomuldivass 28320 grponpcan 28322 ablodivdiv4 28333 vcm 28355 rngonegcl 35207 isdrngo2 35238 |
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