Theorem rngo0cl 34259

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 34250	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ring0cl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ring0cl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 27920	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ran crn 5347  ‘cfv 6127  1^st c1st 7431  GrpOpcgr 27895  GIdcgi 27896  RingOpscrngo 34234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fo 6133  df-fv 6135  df-riota 6871  df-ov 6913  df-1st 7433  df-2nd 7434  df-grpo 27899  df-gid 27900  df-ablo 27951  df-rngo 34235

This theorem is referenced by:  rngolz  34262  rngorz  34263  rngosn6  34266  rngoueqz  34280  rngoidl  34364  0idl  34365  keridl  34372  prnc  34407  isdmn3  34414