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| Mirrors > Home > MPE Home > Th. List > domnring | Structured version Visualization version GIF version | ||
| Description: A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
| Ref | Expression |
|---|---|
| domnring | ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | domnnzr 20706 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
| 2 | nzrring 20516 | . 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Ringcrg 20230 NzRingcnzr 20512 Domncdomn 20692 |
| 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 2708 ax-nul 5306 |
| 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 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-nzr 20513 df-domn 20695 |
| This theorem is referenced by: domneq0 20708 isdomn4 20716 domneq0r 20724 fidomndrnglem 20773 fidomndrng 20774 abvtrivg 20834 domnchr 21547 znidomb 21580 deg1ldgdomn 26133 deg1mul 26154 ply1domn 26163 r1pid2 26201 domnprodn0 33279 r1peuqusdeg1 35648 deg1pow 42142 domnexpgn0cl 42533 fidomncyc 42545 proot1mul 43206 proot1hash 43207 deg1mhm 43212 lidldomn1 48147 uzlidlring 48151 domnmsuppn0 48285 |
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