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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 34164 | . 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | |
2 | prrngorngo 34161 | . . . 4 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | |
3 | iscrngo 34106 | . . . . 5 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
4 | 3 | baibr 983 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
6 | 5 | pm5.32i 672 | . 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
7 | 1, 6 | bitri 264 | 1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 ∈ wcel 2137 RingOpscrngo 34004 Com2ccm2 34099 CRingOpsccring 34103 PrRingcprrng 34156 Dmncdmn 34157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-rex 3054 df-rab 3057 df-v 3340 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-sn 4320 df-pr 4322 df-op 4326 df-uni 4587 df-br 4803 df-iota 6010 df-fv 6055 df-crngo 34104 df-prrngo 34158 df-dmn 34159 |
This theorem is referenced by: dmncrng 34166 flddmn 34168 isdmn3 34184 |
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