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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 35273 | . 2 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 RingOpscrngo 35171 Com2ccm2 35266 CRingOpsccring 35270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3942 df-crngo 35271 |
This theorem is referenced by: crngm23 35279 crngm4 35280 crngohomfo 35283 isidlc 35292 dmnrngo 35334 prnc 35344 isfldidl 35345 isfldidl2 35346 ispridlc 35347 pridlc3 35350 isdmn3 35351 |
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