Theorem isidom 20696

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

Step	Hyp	Ref	Expression
1		df-idom 20667	. 2 ⊢ IDomn = (CRing ∩ Domn)
2	1	elin2 4144	1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  CRingccrg 20209  Domncdomn 20663  IDomncidom 20664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-idom 20667

This theorem is referenced by:  fldidom  20742  fiidomfld  20745  znfld  21553  znidomb  21554  ply1idom  26103  fta1glem1  26146  fta1glem2  26147  fta1g  26148  fta1b  26150  idomrootle  26151  lgsqrlem1  27326  lgsqrlem2  27327  lgsqrlem3  27328  lgsqrlem4  27329  idompropd  33357  subridom  33365  dvdsruasso  33463  qsidomlem1  33530  qsidomlem2  33531  zringidom  33629  r1pid2OLD  33687  idomnnzpownz  42588  idomnnzgmulnz  42589  aks6d1c5lem3  42593  aks6d1c5lem2  42594  deg1gprod  42596  deg1pow  42597  idomodle  43640  proot1mul  43643  proot1hash  43644