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Mirrors > Home > MPE Home > Th. List > iscrng | Structured version Visualization version GIF version |
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 6891 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
2 | ringmgp.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | 1, 2 | eqtr4di 2789 | . . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺) |
4 | 3 | eleq1d 2817 | . 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd)) |
5 | df-cring 20137 | . 2 ⊢ CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd} | |
6 | 4, 5 | elrab2 3686 | 1 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 CMndccmn 19696 mulGrpcmgp 20035 Ringcrg 20134 CRingccrg 20135 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-cring 20137 |
This theorem is referenced by: crngmgp 20142 crngring 20146 iscrng2 20153 crngpropd 20184 iscrngd 20187 prdscrngd 20217 subrgcrng 20473 psrcrng 21844 cntrcrng 32650 |
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