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Theorem nzrring 17783
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
nzrring  |-  ( R  e. NzRing  ->  R  e.  Ring )

Proof of Theorem nzrring
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
2 eqid 2443 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2isnzr 17781 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  ( 0g
`  R ) ) )
43simplbi 460 1  |-  ( R  e. NzRing  ->  R  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1804    =/= wne 2638   ` cfv 5578   0gc0g 14714   1rcur 17027   Ringcrg 17072  NzRingcnzr 17779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-nzr 17780
This theorem is referenced by:  opprnzr  17787  nzrunit  17789  domnring  17819  domnchr  18442  uvcf1  18696  lindfind2  18726  frlmisfrlm  18756  nminvr  21051  deg1pw  22394  ply1nz  22395  ply1remlem  22436  ply1rem  22437  facth1  22438  fta1glem1  22439  fta1glem2  22440  zrhnm  27823  mon1pid  31141  mon1psubm  31142  nzrneg1ne0  32695  islindeps2  32819
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