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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 6538 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
2 | ringmgp.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | 1, 2 | syl6eqr 2849 | . . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺) |
4 | 3 | eleq1d 2867 | . 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd)) |
5 | df-cring 18990 | . 2 ⊢ CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd} | |
6 | 4, 5 | elrab2 3621 | 1 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ‘cfv 6225 CMndccmn 18633 mulGrpcmgp 18929 Ringcrg 18987 CRingccrg 18988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-iota 6189 df-fv 6233 df-cring 18990 |
This theorem is referenced by: crngmgp 18995 crngring 18998 iscrng2 19003 crngpropd 19023 iscrngd 19026 prdscrngd 19053 subrgcrng 19229 psrcrng 19881 cntrcrng 30510 |
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