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Theorem iscrng 20258
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 6907 . . . 4 (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
2 ringmgp.g . . . 4 𝐺 = (mulGrp‘𝑅)
31, 2eqtr4di 2793 . . 3 (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺)
43eleq1d 2824 . 2 (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd))
5 df-cring 20254 . 2 CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd}
64, 5elrab2 3698 1 (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  cfv 6563  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-cring 20254
This theorem is referenced by:  crngmgp  20259  crngring  20263  iscrng2  20270  crngpropd  20303  iscrngd  20306  prdscrngd  20336  subrgcrng  20592  psrcrng  22010  cntrcrng  33056  0ringcring  33239
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