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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 38003 | . 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 2108 RingOpscrngo 37901 Com2ccm2 37996 CRingOpsccring 38000 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-crngo 38001 |
| This theorem is referenced by: crngm23 38009 crngm4 38010 crngohomfo 38013 isidlc 38022 dmnrngo 38064 prnc 38074 isfldidl 38075 isfldidl2 38076 ispridlc 38077 pridlc3 38080 isdmn3 38081 |
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