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Theorem nzrrng 16332
Description: A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
nzrrng  |-  ( R  e. NzRing  ->  R  e.  Ring )

Proof of Theorem nzrrng
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
2 eqid 2436 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2isnzr 16330 . 2  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  ( 1r `  R
)  =/=  ( 0g
`  R ) ) )
43simplbi 447 1  |-  ( R  e. NzRing  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2599   ` cfv 5454   0gc0g 13723   Ringcrg 15660   1rcur 15662  NzRingcnzr 16328
This theorem is referenced by:  opprnzr  16335  nzrunit  16337  domnrng  16356  domnchr  16813  nminvr  18705  deg1pw  20043  ply1nz  20044  ply1remlem  20085  ply1rem  20086  facth1  20087  fta1glem1  20088  fta1glem2  20089  zrhnm  24353  uvcf1  27218  lindfind2  27265  mon1pid  27501  mon1psubm  27502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-nzr 16329
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