Theorem domnring 20911

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 20910	. 2 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
2		nzrring 20294	. 2 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)

Syntax hints:   → wi 4   ∈ wcel 2106  Ringcrg 20055  NzRingcnzr 20290  Domncdomn 20895

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  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 7411  df-nzr 20291  df-domn 20899

This theorem is referenced by:  domneq0  20912  isdomn4  20917  abvn0b  20919  fidomndrnglem  20924  fidomndrng  20925  domnchr  21083  znidomb  21116  deg1ldgdomn  25611  ply1domn  25640  proot1mul  41931  proot1hash  41932  deg1mhm  41939  lidldomn1  46813  uzlidlring  46817  domnmsuppn0  47035