Theorem rngo0cl 38656

Description: Obsolete theorem, use ring0cl 20409 instead. A ring has an additive identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) (Proof modification 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 38647	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ring0cl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ring0cl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30981	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 18	1 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ran crn 5660  ‘cfv 6537  1^st c1st 7987  GrpOpcgr 30956  GIdcgi 30957  RingOpscrngo 38631

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-riota 7373  df-ov 7419  df-1st 7989  df-2nd 7990  df-grpo 30960  df-gid 30961  df-ablo 31012  df-rngo 38632

This theorem is used by:  rngolz  38659  rngorz  38660  rngosn6  38663  rngoueqz  38677  rngoidl  38761  0idl  38762  keridl  38769  prnc  38804  isdmn3  38811