Theorem rngogrpo 38248

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 38246	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp)
3		ablogrpo 30636	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6493  1^st c1st 7934  GrpOpcgr 30578  AbelOpcablo 30633  RingOpscrngo 38232

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 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

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-rab 3391  df-v 3432  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-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7364  df-1st 7936  df-2nd 7937  df-ablo 30634  df-rngo 38233

This theorem is referenced by:  rngone0  38249  rngogcl  38250  rngoaass  38252  rngorcan  38255  rngolcan  38256  rngo0cl  38257  rngo0rid  38258  rngo0lid  38259  rngolz  38260  rngorz  38261  rngosn3  38262  rngonegcl  38265  rngoaddneg1  38266  rngoaddneg2  38267  rngosub  38268  rngodm1dm2  38270  rngorn1  38271  rngonegmn1l  38279  rngonegmn1r  38280  rngogrphom  38309  rngohom0  38310  rngohomsub  38311  rngokerinj  38313  keridl  38370  dmncan1  38414