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Theorem domnrng 17289
Description: A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
Assertion
Ref Expression
domnrng  |-  ( R  e. Domn  ->  R  e.  Ring )

Proof of Theorem domnrng
StepHypRef Expression
1 domnnzr 17288 . 2  |-  ( R  e. Domn  ->  R  e. NzRing )
2 nzrrng 17264 . 2  |-  ( R  e. NzRing  ->  R  e.  Ring )
31, 2syl 16 1  |-  ( R  e. Domn  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1755   Ringcrg 16576  NzRingcnzr 17260  Domncdomn 17272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-nul 4409
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-ov 6083  df-nzr 17261  df-domn 17276
This theorem is referenced by:  domneq0  17290  abvn0b  17295  fidomndrnglem  17299  fidomndrng  17300  domnchr  17804  znidomb  17835  deg1ldgdomn  21449  ply1domn  21479  proot1mul  29406  proot1hash  29410  deg1mhm  29417  domnmsuppn0  30610
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