Theorem rngogrpo 35190

Description: A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage 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 35188	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 28326	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ‘cfv 6357  1^st c1st 7689  GrpOpcgr 28268  AbelOpcablo 28323  RingOpscrngo 35174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-1st 7691  df-2nd 7692  df-ablo 28324  df-rngo 35175

This theorem is referenced by:  rngone0  35191  rngogcl  35192  rngoaass  35194  rngorcan  35197  rngolcan  35198  rngo0cl  35199  rngo0rid  35200  rngo0lid  35201  rngolz  35202  rngorz  35203  rngosn3  35204  rngonegcl  35207  rngoaddneg1  35208  rngoaddneg2  35209  rngosub  35210  rngodm1dm2  35212  rngorn1  35213  rngonegmn1l  35221  rngonegmn1r  35222  rngogrphom  35251  rngohom0  35252  rngohomsub  35253  rngokerinj  35255  keridl  35312  dmncan1  35356