![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 20694 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | nzrring 20660 | . 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 2107 Ringcrg 19894 NzRingcnzr 20656 Domncdomn 20679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-nul 5262 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6444 df-fv 6500 df-ov 7353 df-nzr 20657 df-domn 20683 |
This theorem is referenced by: domneq0 20696 abvn0b 20701 fidomndrnglem 20706 fidomndrng 20707 domnchr 20864 znidomb 20897 deg1ldgdomn 25387 ply1domn 25416 isdomn4 40555 proot1mul 41428 proot1hash 41429 deg1mhm 41436 lidldomn1 46010 uzlidlring 46018 domnmsuppn0 46236 |
Copyright terms: Public domain | W3C validator |