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Mirrors > Home > ILE Home > Th. List > domnnzr | 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 2193 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2193 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2193 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdomn 13743 | . 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 274 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ‘cfv 5246 (class class class)co 5910 Basecbs 12605 .rcmulr 12683 0gc0g 12854 NzRingcnzr 13653 Domncdomn 13730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-cnex 7953 ax-resscn 7954 ax-1re 7956 ax-addrcl 7959 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-iota 5207 df-fun 5248 df-fn 5249 df-fv 5254 df-riota 5865 df-ov 5913 df-inn 8973 df-ndx 12608 df-slot 12609 df-base 12611 df-0g 12856 df-domn 13733 |
This theorem is referenced by: domnring 13745 znidomb 14117 |
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