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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 20744 | . 2 ⊢ (𝜑 → 𝑅 ∈ CRing) |
3 | 2 | crngringd 20264 | 1 ⊢ (𝜑 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Ringcrg 20251 IDomncidom 20710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-cring 20254 df-idom 20713 |
This theorem is referenced by: fracfld 33290 dvdsruasso 33393 dvdsruasso2 33394 unitpidl1 33432 mxidlirredi 33479 mxidlirred 33480 rsprprmprmidlb 33531 rprmasso 33533 rprmasso2 33534 unitmulrprm 33536 rprmirred 33539 rprmirredb 33540 1arithidomlem1 33543 1arithidomlem2 33544 1arithidom 33545 pidufd 33551 1arithufdlem2 33553 1arithufdlem4 33555 dfufd2lem 33557 dfufd2 33558 r1pid2OLD 33609 assafld 33665 algextdeglem7 33729 idomnnzpownz 42114 deg1gprod 42122 deg1pow 42123 aks6d1c6lem3 42154 unitscyglem5 42181 |
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