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| Mirrors > Home > MPE Home > Th. List > idomringd | Structured version Visualization version GIF version | ||
| Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| Ref | Expression |
|---|---|
| idomringd.1 | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
| Ref | Expression |
|---|---|
| idomringd | ⊢ (𝜑 → 𝑅 ∈ Ring) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | idomringd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
| 2 | 1 | idomcringd 20727 | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) |
| 3 | 2 | crngringd 20243 | 1 ⊢ (𝜑 → 𝑅 ∈ Ring) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Ringcrg 20230 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-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-cring 20233 df-idom 20696 |
| This theorem is referenced by: fracfld 33310 dvdsruasso 33413 dvdsruasso2 33414 unitpidl1 33452 mxidlirredi 33499 mxidlirred 33500 rsprprmprmidlb 33551 rprmasso 33553 rprmasso2 33554 unitmulrprm 33556 rprmirred 33559 rprmirredb 33560 1arithidomlem1 33563 1arithidomlem2 33564 1arithidom 33565 pidufd 33571 1arithufdlem2 33573 1arithufdlem4 33575 dfufd2lem 33577 dfufd2 33578 r1pid2OLD 33629 assafld 33688 fldextrspunlem1 33725 algextdeglem7 33764 idomnnzpownz 42133 deg1gprod 42141 deg1pow 42142 aks6d1c6lem3 42173 unitscyglem5 42200 |
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