![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2725 | . . 3 ⊢ (GId‘𝐺) = (GId‘𝐺) | |
3 | grpinvcl.2 | . . 3 ⊢ 𝑁 = (inv‘𝐺) | |
4 | 1, 2, 3 | grpoinvval 30405 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (℩𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))) |
5 | 1, 2 | grpoinveu 30401 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) |
6 | riotacl 7393 | . . 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 2825 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃!wreu 3361 ran crn 5679 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 GrpOpcgr 30371 GIdcgi 30372 invcgn 30373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-grpo 30375 df-gid 30376 df-ginv 30377 |
This theorem is referenced by: grpoinvid1 30410 grpoinvid2 30411 grpolcan 30412 grpo2inv 30413 grpoinvf 30414 grpoinvop 30415 grpodivinv 30418 grpoinvdiv 30419 grpodivf 30420 grpomuldivass 30423 grponpcan 30425 ablodivdiv4 30436 vcm 30458 rngonegcl 37528 isdrngo2 37559 |
Copyright terms: Public domain | W3C validator |