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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 35894 | . 2 ⊢ CRingOps = (RingOps ∩ Com2) | |
2 | 1 | elin2 4116 | 1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2110 RingOpscrngo 35794 Com2ccm2 35889 CRingOpsccring 35893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3415 df-in 3878 df-crngo 35894 |
This theorem is referenced by: iscrngo2 35897 iscringd 35898 crngorngo 35900 fldcrng 35904 isfld2 35905 isdmn2 35955 |
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