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Mirrors > Home > MPE Home > Th. List > domnnzr | Structured version Visualization version GIF version |
Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
Ref | Expression |
---|---|
domnnzr | ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2731 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdomn 21114 | . 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))))) |
5 | 4 | simplbi 497 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 .rcmulr 17205 0gc0g 17392 NzRingcnzr 20407 Domncdomn 21100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-domn 21104 |
This theorem is referenced by: domnring 21116 isdomn4 21122 opprdomn 21123 abvn0b 21124 fidomndrng 21130 domnchr 21307 znidomb 21340 nrgdomn 24421 ply1domn 25890 fta1glem1 25932 fta1glem2 25933 fta1b 25936 lgsqrlem4 27103 qsidomlem1 32860 idomrootle 42252 deg1mhm 42264 |
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