Theorem rngo0cl 38231

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 38222	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ring0cl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ring0cl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30574	. 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 5623  ‘cfv 6490  1^st c1st 7931  GrpOpcgr 30549  GIdcgi 30550  RingOpscrngo 38206

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 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

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 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-riota 7315  df-ov 7361  df-1st 7933  df-2nd 7934  df-grpo 30553  df-gid 30554  df-ablo 30605  df-rngo 38207

This theorem is referenced by:  rngolz  38234  rngorz  38235  rngosn6  38238  rngoueqz  38252  rngoidl  38336  0idl  38337  keridl  38344  prnc  38379  isdmn3  38386