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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmncrng | Structured version Visualization version GIF version |
Description: A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
dmncrng | ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isdmn2 36911 | . 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 CRingOpsccring 36849 PrRingcprrng 36902 Dmncdmn 36903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-crngo 36850 df-prrngo 36904 df-dmn 36905 |
This theorem is referenced by: dmnrngo 36913 dmncan2 36933 |
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