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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 2821 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
3 | grpinvcl.2 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinvval 28299 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))) |
5 | 1, 2 | grpoinveu 28295 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) |
6 | riotacl 7130 | . . 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 2913 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 ran crn 5555 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 GrpOpcgr 28265 GIdcgi 28266 invcgn 28267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-grpo 28269 df-gid 28270 df-ginv 28271 |
This theorem is referenced by: grpoinvid1 28304 grpoinvid2 28305 grpolcan 28306 grpo2inv 28307 grpoinvf 28308 grpoinvop 28309 grpodivinv 28312 grpoinvdiv 28313 grpodivf 28314 grpomuldivass 28317 grponpcan 28319 ablodivdiv4 28330 vcm 28352 rngonegcl 35204 isdrngo2 35235 |
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