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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 20723 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | nzrring 20533 | . 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 2106 Ringcrg 20251 NzRingcnzr 20529 Domncdomn 20709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-nzr 20530 df-domn 20712 |
This theorem is referenced by: domneq0 20725 isdomn4 20733 domneq0r 20741 fidomndrnglem 20790 fidomndrng 20791 abvtrivg 20851 domnchr 21565 znidomb 21598 deg1ldgdomn 26148 deg1mul 26169 ply1domn 26178 r1pid2 26216 domnprodn0 33262 r1peuqusdeg1 35628 deg1pow 42123 domnexpgn0cl 42510 fidomncyc 42522 proot1mul 43183 proot1hash 43184 deg1mhm 43189 lidldomn1 48075 uzlidlring 48079 domnmsuppn0 48214 |
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