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Mathbox for Jeff Madsen |
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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 36853 | . 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
2 | 1 | simplbi 499 | 1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 RingOpscrngo 36751 Com2ccm2 36846 CRingOpsccring 36850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3955 df-crngo 36851 |
This theorem is referenced by: crngm23 36859 crngm4 36860 crngohomfo 36863 isidlc 36872 dmnrngo 36914 prnc 36924 isfldidl 36925 isfldidl2 36926 ispridlc 36927 pridlc3 36930 isdmn3 36931 |
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