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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 30544 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥 ∈ 𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈))) |
4 | simplr 769 | . 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 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ran crn 5690 ‘cfv 6563 (class class class)co 7431 GrpOpcgr 30518 GIdcgi 30519 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-riota 7388 df-ov 7434 df-grpo 30522 df-gid 30523 |
This theorem is referenced by: grporcan 30547 grpoinvid1 30557 grpoinvid2 30558 grponpcan 30572 vc0rid 30602 vcm 30605 nv0rid 30664 rngo0rid 37907 rngolz 37909 |
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