MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isidom Structured version   Visualization version   GIF version

Theorem isidom 19236
Description: An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
Assertion
Ref Expression
isidom (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Proof of Theorem isidom
StepHypRef Expression
1 df-idom 19217 . 2 IDomn = (CRing ∩ Domn)
21elin2 3784 1 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  CRingccrg 18480  Domncdomn 19212  IDomncidom 19213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-idom 19217
This theorem is referenced by:  fldidom  19237  fiidomfld  19240  znfld  19841  znidomb  19842  recvs  22869  ply1idom  23805  fta1glem1  23846  fta1glem2  23847  fta1g  23848  fta1b  23850  lgsqrlem1  24988  lgsqrlem2  24989  lgsqrlem3  24990  lgsqrlem4  24991  idomrootle  37289  idomodle  37290  proot1mul  37293  proot1hash  37294
  Copyright terms: Public domain W3C validator