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Mirrors > Home > MPE Home > Th. List > Mathboxes > crngorngo | Structured version Visualization version GIF version |
Description: A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
crngorngo | ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscrngo 37982 | . 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 RingOpscrngo 37880 Com2ccm2 37975 CRingOpsccring 37979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-in 3969 df-crngo 37980 |
This theorem is referenced by: crngm23 37988 crngm4 37989 crngohomfo 37992 isidlc 38001 dmnrngo 38043 prnc 38053 isfldidl 38054 isfldidl2 38055 ispridlc 38056 pridlc3 38059 isdmn3 38060 |
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