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| Description: A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| ringmgp.g | ⊢ 𝐺 = (mulGrp‘𝑅) | 
| Ref | Expression | 
|---|---|
| iscrng | ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6905 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
| 2 | ringmgp.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 3 | 1, 2 | eqtr4di 2794 | . . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺) | 
| 4 | 3 | eleq1d 2825 | . 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd)) | 
| 5 | df-cring 20234 | . 2 ⊢ CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd} | |
| 6 | 4, 5 | elrab2 3694 | 1 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 CMndccmn 19799 mulGrpcmgp 20138 Ringcrg 20231 CRingccrg 20232 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-cring 20234 | 
| This theorem is referenced by: crngmgp 20239 crngring 20243 iscrng2 20250 crngpropd 20287 iscrngd 20290 prdscrngd 20320 subrgcrng 20576 psrcrng 21993 cntrcrng 33074 0ringcring 33257 | 
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