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Mathbox for Jeff Madsen |
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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 34121 | . 2 ⊢ CRingOps = (RingOps ∩ Com2) | |
2 | 1 | elin2 3952 | 1 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 ∈ wcel 2145 RingOpscrngo 34021 Com2ccm2 34116 CRingOpsccring 34120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-in 3730 df-crngo 34121 |
This theorem is referenced by: iscrngo2 34124 iscringd 34125 crngorngo 34127 fldcrng 34131 isfld2 34132 isdmn2 34182 |
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