Theorem ringgrpd 20164

Description: A ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
ringgrpd.1	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
ringgrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem ringgrpd

Step	Hyp	Ref	Expression
1		ringgrpd.1	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		ringgrp 20160	. 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2113  Grpcgrp 18850  Ringcrg 20155

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5248

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-ring 20157

This theorem is referenced by:  crnggrpd  20169  ringcom  20202  lringuplu  20463  isdomn4  20635  drnggrpd  20657  lssvnegcl  20893  rngqiprngimfo  21242  rngqiprngfulem4  21255  ofldchr  21517  asclmulg  21843  psrdi  21905  psrdir  21906  evlslem1  22020  mhplss  22073  psdmvr  22087  evls1addd  22289  evls1maprhm  22294  rhmcomulmpl  22300  rhmmpl  22301  r1pid2  26097  ringdi22  33207  elrgspnlem1  33218  elrgspnlem2  33219  elrgspnlem4  33221  elrgspn  33222  erler  33241  rlocmulval  33245  rloccring  33246  fracfld  33283  znfermltl  33340  qsdrngilem  33468  qsdrngi  33469  qsdrnglem2  33470  qsdrng  33471  evls1subd  33544  q1pdir  33572  r1pcyc  33576  r1padd1  33577  r1pid2OLD  33578  r1plmhm  33579  r1pquslmic  33580  mplmulmvr  33592  mplvrpmmhm  33596  esplyfval2  33607  esplyfval3  33614  esplyind  33615  assalactf1o  33671  irredminply  33752  algextdeglem8  33760  rtelextdg2lem  33762  2sqr3minply  33816  cos9thpiminplylem6  33823  cos9thpiminply  33824  zrhcntr  34015  ellcsrspsn  35708  ply1divalg3  35709  r1peuqusdeg1  35710  fldhmf1  42206  aks6d1c1p2  42225  aks6d1c5lem3  42253  aks5lem2  42303  aks5lem5a  42307  rhmcomulpsr  42672  rhmpsr  42673  evlsmaprhm  42691