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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 19295 | . 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | nzrring 19261 | . 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 1990 Ringcrg 18547 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-ne 2795 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-nzr 19258 df-domn 19284 |
This theorem is referenced by: domneq0 19297 abvn0b 19302 fidomndrnglem 19306 fidomndrng 19307 domnchr 19880 znidomb 19910 deg1ldgdomn 23854 ply1domn 23883 proot1mul 37777 proot1hash 37778 deg1mhm 37785 lidldomn1 41921 uzlidlring 41929 domnmsuppn0 42150 |
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