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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iscrngo | Structured version Visualization version GIF version | ||
| Description: The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.) |
| Ref | Expression |
|---|---|
| iscrngo | ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-crngo 38001 | . 2 ⊢ CRingOps = (RingOps ∩ Com2) | |
| 2 | 1 | elin2 4203 | 1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ 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: iscrngo2 38004 iscringd 38005 crngorngo 38007 fldcrngo 38011 isfld2 38012 isdmn2 38062 |
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