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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 6191 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
2 | ringmgp.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | 1, 2 | syl6eqr 2674 | . . 3 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝐺) |
4 | 3 | eleq1d 2686 | . 2 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ CMnd ↔ 𝐺 ∈ CMnd)) |
5 | df-cring 18550 | . 2 ⊢ CRing = {𝑟 ∈ Ring ∣ (mulGrp‘𝑟) ∈ CMnd} | |
6 | 4, 5 | elrab2 3366 | 1 ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 CMndccmn 18193 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-cring 18550 |
This theorem is referenced by: crngmgp 18555 crngring 18558 iscrng2 18563 crngpropd 18583 iscrngd 18586 prdscrngd 18613 subrgcrng 18784 psrcrng 19413 |
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