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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 2622 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2622 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2622 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdomn 19294 | . 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 476 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 0gc0g 16100 NzRingcnzr 19257 Domncdomn 19280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-domn 19284 |
This theorem is referenced by: domnring 19296 opprdomn 19301 abvn0b 19302 fidomndrng 19307 domnchr 19880 znidomb 19910 nrgdomn 22475 ply1domn 23883 fta1glem1 23925 fta1glem2 23926 fta1b 23929 lgsqrlem4 25074 idomrootle 37773 deg1mhm 37785 |
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