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Theorem iscrng 18486
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.
StepHypRef Expression
1 fveq2 6153 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 ringmgp.g . . . 4 𝐺 = (mulGrp‘𝑅)
31, 2syl6eqr 2673 . . 3 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2683 . 2 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5 df-cring 18482 . 2 CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
64, 5elrab2 3352 1 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  cfv 5852  CMndccmn 18125  mulGrpcmgp 18421  Ringcrg 18479  CRingccrg 18480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-cring 18482
This theorem is referenced by:  crngmgp  18487  crngring  18490  iscrng2  18495  crngpropd  18515  iscrngd  18518  prdscrngd  18545  subrgcrng  18716  psrcrng  19345
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