Theorem rngo0cl 38240

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 38231	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ring0cl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ring0cl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30585	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ran crn 5632  ‘cfv 6499  1^st c1st 7940  GrpOpcgr 30560  GIdcgi 30561  RingOpscrngo 38215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5376  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-riota 7324  df-ov 7370  df-1st 7942  df-2nd 7943  df-grpo 30564  df-gid 30565  df-ablo 30616  df-rngo 38216

This theorem is referenced by:  rngolz  38243  rngorz  38244  rngosn6  38247  rngoueqz  38261  rngoidl  38345  0idl  38346  keridl  38353  prnc  38388  isdmn3  38395