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Theorem idomringd 20631
Description: An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypothesis
Ref Expression
idomringd.1 (𝜑𝑅 ∈ IDomn)
Assertion
Ref Expression
idomringd (𝜑𝑅 ∈ Ring)

Proof of Theorem idomringd
StepHypRef Expression
1 idomringd.1 . . 3 (𝜑𝑅 ∈ IDomn)
21idomcringd 20630 . 2 (𝜑𝑅 ∈ CRing)
32crngringd 20149 1 (𝜑𝑅 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  Ringcrg 20136  IDomncidom 20596
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 2008  ax-8 2111  ax-9 2119  ax-ext 2701
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-cring 20139  df-idom 20599
This theorem is referenced by:  fracfld  33260  dvdsruasso  33335  dvdsruasso2  33336  unitpidl1  33374  mxidlirredi  33421  mxidlirred  33422  rsprprmprmidlb  33473  rprmasso  33475  rprmasso2  33476  unitmulrprm  33478  rprmirred  33481  rprmirredb  33482  1arithidomlem1  33485  1arithidomlem2  33486  1arithidom  33487  pidufd  33493  1arithufdlem2  33495  1arithufdlem4  33497  dfufd2lem  33499  dfufd2  33500  r1pid2OLD  33553  assafld  33612  fldextrspunlem1  33649  algextdeglem7  33692  idomnnzpownz  42108  deg1gprod  42116  deg1pow  42117  aks6d1c6lem3  42148  unitscyglem5  42175
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