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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 20061 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | nzrring 20027 | . 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 2111 Ringcrg 19290 NzRingcnzr 20023 Domncdomn 20046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-nzr 20024 df-domn 20050 |
This theorem is referenced by: domneq0 20063 abvn0b 20068 fidomndrnglem 20072 fidomndrng 20073 domnchr 20224 znidomb 20253 deg1ldgdomn 24695 ply1domn 24724 proot1mul 40143 proot1hash 40144 deg1mhm 40151 lidldomn1 44545 uzlidlring 44553 domnmsuppn0 44771 |
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