Theorem iscrng 20473

Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypothesis

Ref	Expression
ringmgp.g	⊢ 𝐺 = (mulGrp‘𝑅)

Assertion

Ref	Expression
iscrng	⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))

Proof of Theorem iscrng

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2		ringmgp.g	. . . 4 ⊢ 𝐺 = (mulGrp‘𝑅)
3	1, 2	eqtr4di 2813	. . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
4	3	eleq1d 2845	. 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5		df-cring 20469	. 2 ⊢ CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
6	4, 5	elrab2 3648	1 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ‘cfv 6541  CMndccmn 20001  mulGrpcmgp 20367  Ringcrg 20466  CRingccrg 20467

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-ext 2732

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-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  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-iota 6496  df-fv 6549  df-cring 20469

This theorem is used by:  crngmgp  20474  crngring  20479  iscrng2  20486  crngpropd  20527  iscrngd  20530  prdscrngd  20558  subrgcrng  20834  psrcrng  22286  cntrcrng  33690  0ringcring  33861