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Theorem nzrrng 17476
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 2454 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
2 eqid 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
31, 2isnzr 17474 . 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 1758    =/= wne 2648   ` cfv 5529   0gc0g 14501   1rcur 16735   Ringcrg 16778  NzRingcnzr 17472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-nzr 17473
This theorem is referenced by:  opprnzr  17479  nzrunit  17481  domnrng  17501  domnchr  18098  uvcf1  18352  lindfind2  18382  frlmisfrlm  18412  nminvr  20392  deg1pw  21735  ply1nz  21736  ply1remlem  21777  ply1rem  21778  facth1  21779  fta1glem1  21780  fta1glem2  21781  zrhnm  26566  mon1pid  29744  mon1psubm  29745  nzrneg1ne0  30951  islindeps2  31172
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