Theorem domnnzr 19295

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

Assertion

Ref	Expression
domnnzr	|- ( R e. Domn -> R e. NzRing )

Proof of Theorem domnnzr

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` R ) = ( Base ` R )
2		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
3		eqid 2622	. . 3 |- ( 0g ` R ) = ( 0g ` R )
4	1, 2, 3	isdomn 19294	. 2 |- ( R e. Domn <-> ( R e. NzRing /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = ( 0g ` R ) -> ( x = ( 0g ` R ) \/ y = ( 0g ` R ) ) ) ) )
5	4	simplbi 476	1 |- ( R e. Domn -> R e. NzRing )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942   0gc0g 16100  NzRingcnzr 19257  Domncdomn 19280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-domn 19284

This theorem is referenced by:  domnring  19296  opprdomn  19301  abvn0b  19302  fidomndrng  19307  domnchr  19880  znidomb  19910  nrgdomn  22475  ply1domn  23883  fta1glem1  23925  fta1glem2  23926  fta1b  23929  lgsqrlem4  25074  idomrootle  37773  deg1mhm  37785