Theorem domnring 21202

Description: A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)

Assertion

Ref	Expression
domnring	⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)

Proof of Theorem domnring

Step	Hyp	Ref	Expression
1		domnnzr 21201	. 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		nzrring 20414	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2105  Ringcrg 20134  NzRingcnzr 20410  Domncdomn 21186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-nzr 20411  df-domn 21190

This theorem is referenced by:  domneq0  21203  isdomn4  21208  abvn0b  21210  fidomndrnglem  21215  fidomndrng  21216  domnchr  21395  znidomb  21428  deg1ldgdomn  25951  ply1domn  25980  proot1mul  42407  proot1hash  42408  deg1mhm  42415  lidldomn1  47071  uzlidlring  47075  domnmsuppn0  47211