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Theorem mon1pn0 20022
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 2404 . . 3  |-  ( Base `  P )  =  (
Base `  P )
3 uc1pn0.z . . 3  |-  .0.  =  ( 0g `  P )
4 eqid 2404 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pn0.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2404 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 20018 . 2  |-  ( F  e.  M  <->  ( F  e.  ( Base `  P
)  /\  F  =/=  .0.  /\  ( (coe1 `  F
) `  ( ( deg1  `  R ) `  F
) )  =  ( 1r `  R ) ) )
87simp2bi 973 1  |-  ( F  e.  M  ->  F  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413   Basecbs 13424   0gc0g 13678   1rcur 15617  Poly1cpl1 16526  coe1cco1 16529   deg1 cdg1 19930  Monic1pcmn1 20001
This theorem is referenced by:  mon1puc1p  20026  deg1submon1p  20028  mon1psubm  27393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-slot 13428  df-base 13429  df-mon1 20006
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