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Theorem rngo0cl 34049
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 = (1st𝑅)
ring0cl.2 𝑋 = ran 𝐺
ring0cl.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngo0cl (𝑅 ∈ RingOps → 𝑍𝑋)

Proof of Theorem rngo0cl
StepHypRef Expression
1 ring0cl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 34040 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ring0cl.2 . . 3 𝑋 = ran 𝐺
4 ring0cl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 27698 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑅 ∈ RingOps → 𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  ran crn 5267  cfv 6049  1st c1st 7332  GrpOpcgr 27673  GIdcgi 27674  RingOpscrngo 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-riota 6775  df-ov 6817  df-1st 7334  df-2nd 7335  df-grpo 27677  df-gid 27678  df-ablo 27729  df-rngo 34025
This theorem is referenced by:  rngolz  34052  rngorz  34053  rngosn6  34056  rngoueqz  34070  rngoidl  34154  0idl  34155  keridl  34162  prnc  34197  isdmn3  34204
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