Theorem isidom 20634

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 20605	. 2 ⊢ IDomn = (CRing ∩ Domn)
2	1	elin2 4166	1 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  CRingccrg 20143  Domncdomn 20601  IDomncidom 20602

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-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-in 3921  df-idom 20605

This theorem is referenced by:  fldidom  20680  fiidomfld  20683  znfld  21470  znidomb  21471  ply1idom  26030  fta1glem1  26073  fta1glem2  26074  fta1g  26075  fta1b  26077  idomrootle  26078  lgsqrlem1  27257  lgsqrlem2  27258  lgsqrlem3  27259  lgsqrlem4  27260  idompropd  33228  subridom  33236  dvdsruasso  33356  qsidomlem1  33423  qsidomlem2  33424  zringidom  33522  r1pid2OLD  33574  idomnnzpownz  42120  idomnnzgmulnz  42121  aks6d1c5lem3  42125  aks6d1c5lem2  42126  deg1gprod  42128  deg1pow  42129  idomodle  43180  proot1mul  43183  proot1hash  43184