Theorem rngo0cl 37103

Description: A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ring0cl.1	⊢ 𝐺 = (1st ‘𝑅)
ring0cl.2	⊢ 𝑋 = ran 𝐺
ring0cl.3	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
rngo0cl	⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

Proof of Theorem rngo0cl

Step	Hyp	Ref	Expression
1		ring0cl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 37094	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ring0cl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ring0cl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30049	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ran crn 5677  ‘cfv 6543  1^st c1st 7977  GrpOpcgr 30024  GIdcgi 30025  RingOpscrngo 37078

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7368  df-ov 7415  df-1st 7979  df-2nd 7980  df-grpo 30028  df-gid 30029  df-ablo 30080  df-rngo 37079

This theorem is referenced by:  rngolz  37106  rngorz  37107  rngosn6  37110  rngoueqz  37124  rngoidl  37208  0idl  37209  keridl  37216  prnc  37251  isdmn3  37258