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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 2825 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2825 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2825 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdomn 19655 | . 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 493 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 .rcmulr 16306 0gc0g 16453 NzRingcnzr 19618 Domncdomn 19641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-domn 19645 |
This theorem is referenced by: domnring 19657 opprdomn 19662 abvn0b 19663 fidomndrng 19668 domnchr 20240 znidomb 20269 nrgdomn 22845 ply1domn 24282 fta1glem1 24324 fta1glem2 24325 fta1b 24328 lgsqrlem4 25487 idomrootle 38616 deg1mhm 38628 |
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