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Mirrors > Home > MPE Home > Th. List > grporid | Structured version Visualization version GIF version |
Description: The identity element of a group is a right identity. (Contributed by NM, 24-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grpoidval.1 | ⊢ 𝑋 = ran 𝐺 |
grpoidval.2 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
grporid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑈) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpoidval.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | grpoidval.2 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | grpoidinv2 27710 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥 ∈ 𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈))) |
4 | simplr 746 | . 2 ⊢ ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥 ∈ 𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈)) → (𝐴𝐺𝑈) = 𝐴) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑈) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 ran crn 5251 ‘cfv 6032 (class class class)co 6794 GrpOpcgr 27684 GIdcgi 27685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-fo 6038 df-fv 6040 df-riota 6755 df-ov 6797 df-grpo 27688 df-gid 27689 |
This theorem is referenced by: grporcan 27713 grpoinvid1 27723 grpoinvid2 27724 grponpcan 27738 vc0rid 27769 vcm 27772 nv0rid 27831 rngo0rid 34052 rngolz 34054 |
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