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Theorem mon1pn0 20074
Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pn0.p  |-  P  =  (Poly1 `  R )
uc1pn0.z  |-  .0.  =  ( 0g `  P )
mon1pn0.m  |-  M  =  (Monic1p `  R )
Assertion
Ref Expression
mon1pn0  |-  ( F  e.  M  ->  F  =/=  .0.  )

Proof of Theorem mon1pn0
StepHypRef Expression
1 uc1pn0.p . . 3  |-  P  =  (Poly1 `  R )
2 eqid 2438 . . 3  |-  ( Base `  P )  =  (
Base `  P )
3 uc1pn0.z . . 3  |-  .0.  =  ( 0g `  P )
4 eqid 2438 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pn0.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2438 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 20070 . 2  |-  ( F  e.  M  <->  ( F  e.  ( Base `  P
)  /\  F  =/=  .0.  /\  ( (coe1 `  F
) `  ( ( deg1  `  R ) `  F
) )  =  ( 1r `  R ) ) )
87simp2bi 974 1  |-  ( F  e.  M  ->  F  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5457   Basecbs 13474   0gc0g 13728   1rcur 15667  Poly1cpl1 16576  coe1cco1 16579   deg1 cdg1 19982  Monic1pcmn1 20053
This theorem is referenced by:  mon1puc1p  20078  deg1submon1p  20080  mon1psubm  27516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-slot 13478  df-base 13479  df-mon1 20058
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