Theorem rngogrpo 38981

Description: Obsolete theorem, use ringgrp 20473 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 38979	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 31158	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 18	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ‘cfv 6543  1^st c1st 8003  GrpOpcgr 31100  AbelOpcablo 31155  RingOpscrngo 38965

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7427  df-1st 8005  df-2nd 8006  df-ablo 31156  df-rngo 38966

This theorem is used by:  rngone0  38982  rngogcl  38983  rngoaass  38985  rngorcan  38988  rngolcan  38989  rngo0cl  38990  rngo0rid  38991  rngo0lid  38992  rngolz  38993  rngorz  38994  rngosn3  38995  rngonegcl  38998  rngoaddneg1  38999  rngoaddneg2  39000  rngosub  39001  rngodm1dm2  39003  rngorn1  39004  rngonegmn1l  39012  rngonegmn1r  39013  rngogrphom  39042  rngohom0  39043  rngohomsub  39044  rngokerinj  39046  keridl  39103  dmncan1  39147