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Mirrors > Home > MPE Home > Th. List > grpolid | Structured version Visualization version GIF version |
Description: The identity element of a group is a left 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 |
---|---|
grpolid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpoidval.1 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
2 | grpoidval.2 | . . 3 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | grpoidinv2 28298 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))) |
4 | 3 | simplld 767 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ran crn 5520 ‘cfv 6324 (class class class)co 7135 GrpOpcgr 28272 GIdcgi 28273 |
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-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-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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-riota 7093 df-ov 7138 df-grpo 28276 df-gid 28277 |
This theorem is referenced by: grpoid 28303 grpoinvid1 28311 grpoinvid2 28312 grpolcan 28313 grpoinvop 28316 ablonncan 28339 vcm 28359 nv0lid 28419 hhssabloilem 29044 grpoeqdivid 35319 ghomidOLD 35327 rngo0lid 35359 rngolz 35360 rngorz 35361 keridl 35470 |
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