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| Mirrors > Home > MPE Home > Th. List > isidom | Structured version Visualization version GIF version | ||
| Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.) |
| Ref | Expression |
|---|---|
| isidom | ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-idom 20696 | . 2 ⊢ IDomn = (CRing ∩ Domn) | |
| 2 | 1 | elin2 4203 | 1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 CRingccrg 20231 Domncdomn 20692 IDomncidom 20693 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-idom 20696 |
| This theorem is referenced by: fldidom 20771 fldidomOLD 20772 fiidomfld 20775 znfld 21579 znidomb 21580 recvsOLD 25180 ply1idom 26164 fta1glem1 26207 fta1glem2 26208 fta1g 26209 fta1b 26211 idomrootle 26212 lgsqrlem1 27390 lgsqrlem2 27391 lgsqrlem3 27392 lgsqrlem4 27393 idompropd 33281 subridom 33289 dvdsruasso 33413 qsidomlem1 33480 qsidomlem2 33481 zringidom 33579 r1pid2OLD 33629 idomnnzpownz 42133 idomnnzgmulnz 42134 aks6d1c5lem3 42138 aks6d1c5lem2 42139 deg1gprod 42141 deg1pow 42142 idomodle 43203 proot1mul 43206 proot1hash 43207 |
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