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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isdmn2 | Structured version Visualization version GIF version |
Description: The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
isdmn2 | ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isdmn 38039 | . 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | |
2 | prrngorngo 38036 | . . . 4 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | |
3 | iscrngo 37981 | . . . . 5 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
4 | 3 | baibr 536 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
6 | 5 | pm5.32i 574 | . 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
7 | 1, 6 | bitri 275 | 1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 RingOpscrngo 37879 Com2ccm2 37974 CRingOpsccring 37978 PrRingcprrng 38031 Dmncdmn 38032 |
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 2707 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-iota 6512 df-fv 6567 df-crngo 37979 df-prrngo 38033 df-dmn 38034 |
This theorem is referenced by: dmncrng 38041 flddmn 38043 isdmn3 38059 |
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