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Theorem grpolid 26754

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.)

**Hypotheses**
Ref	Expression
grpoidval.1	⊢ 𝑋 = ran 𝐺
grpoidval.2	⊢ 𝑈 = (GId‘𝐺)

**Assertion**
Ref	Expression
grpolid	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)

Proof of Theorem grpolid Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoidval.1	. . 3 ⊢ 𝑋 = ran 𝐺
2		grpoidval.2	. . 3 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidinv2 26753	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4	3	simplld 787	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ran crn 5039 ‘cfv 5804 (class class class)co 6549 GrpOpcgr 26727 GIdcgi 26728

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-grpo 26731 df-gid 26732

This theorem is referenced by: grpoid 26758 grpoinvid1 26766 grpoinvid2 26767 grpolcan 26768 grpoinvop 26771 ablonncan 26795 vcm 26815 nv0lid 26875 hhssabloilem 27502 grpoeqdivid 32850 ghomidOLD 32858 rngo0lid 32890 rngolz 32891 rngorz 32892 keridl 33001

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