Theorem isidom 20691

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 20662	. 2 ⊢ IDomn = (CRing ∩ Domn)
2	1	elin2 4144	1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  CRingccrg 20204  Domncdomn 20658  IDomncidom 20659

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 20662

This theorem is referenced by:  fldidom  20737  fiidomfld  20740  znfld  21548  znidomb  21549  ply1idom  26102  fta1glem1  26145  fta1glem2  26146  fta1g  26147  fta1b  26149  idomrootle  26150  lgsqrlem1  27328  lgsqrlem2  27329  lgsqrlem3  27330  lgsqrlem4  27331  idompropd  33359  subridom  33367  dvdsruasso  33465  qsidomlem1  33532  qsidomlem2  33533  zringidom  33631  r1pid2OLD  33689  idomnnzpownz  42582  idomnnzgmulnz  42583  aks6d1c5lem3  42587  aks6d1c5lem2  42588  deg1gprod  42590  deg1pow  42591  idomodle  43634  proot1mul  43637  proot1hash  43638