Theorem grporid 28296

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.)

Hypotheses

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

Assertion

Ref	Expression
grporid	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑈) = 𝐴)

Proof of Theorem grporid

Dummy variable 𝑥 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 28294	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑥 ∈ 𝑋 ((𝑥𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑥) = 𝑈)))
4		simplr 767	. 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)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  ran crn 5558  ‘cfv 6357  (class class class)co 7158  GrpOpcgr 28268  GIdcgi 28269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-riota 7116  df-ov 7161  df-grpo 28272  df-gid 28273

This theorem is referenced by:  grporcan  28297  grpoinvid1  28307  grpoinvid2  28308  grponpcan  28322  vc0rid  28352  vcm  28355  nv0rid  28414  rngo0rid  35200  rngolz  35202