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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 35336 | . 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | |
2 | prrngorngo 35333 | . . . 4 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | |
3 | iscrngo 35278 | . . . . 5 ⊢ (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2)) | |
4 | 3 | baibr 539 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
5 | 2, 4 | syl 17 | . . 3 ⊢ (𝑅 ∈ PrRing → (𝑅 ∈ Com2 ↔ 𝑅 ∈ CRingOps)) |
6 | 5 | pm5.32i 577 | . 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2) ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
7 | 1, 6 | bitri 277 | 1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 RingOpscrngo 35176 Com2ccm2 35271 CRingOpsccring 35275 PrRingcprrng 35328 Dmncdmn 35329 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-crngo 35276 df-prrngo 35330 df-dmn 35331 |
This theorem is referenced by: dmncrng 35338 flddmn 35340 isdmn3 35356 |
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