Theorem rngogrpo 38660

Description: Obsolete theorem, use ringgrp 20377 instead. A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ringgrp.1	⊢ 𝐺 = (1st ‘𝑅)

Assertion

Ref	Expression
rngogrpo	⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Proof of Theorem rngogrpo

Step	Hyp	Ref	Expression
1		ringgrp.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngoablo 38658	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 31028	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 18	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  1^st c1st 7984  GrpOpcgr 30970  AbelOpcablo 31025  RingOpscrngo 38644

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 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-1st 7986  df-2nd 7987  df-ablo 31026  df-rngo 38645

This theorem is used by:  rngone0  38661  rngogcl  38662  rngoaass  38664  rngorcan  38667  rngolcan  38668  rngo0cl  38669  rngo0rid  38670  rngo0lid  38671  rngolz  38672  rngorz  38673  rngosn3  38674  rngonegcl  38677  rngoaddneg1  38678  rngoaddneg2  38679  rngosub  38680  rngodm1dm2  38682  rngorn1  38683  rngonegmn1l  38691  rngonegmn1r  38692  rngogrphom  38721  rngohom0  38722  rngohomsub  38723  rngokerinj  38725  keridl  38782  dmncan1  38826