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Theorem mon1pn0 19636
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 2358 . . 3  |-  ( Base `  P )  =  (
Base `  P )
3 uc1pn0.z . . 3  |-  .0.  =  ( 0g `  P )
4 eqid 2358 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pn0.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2358 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 19632 . 2  |-  ( F  e.  M  <->  ( F  e.  ( Base `  P
)  /\  F  =/=  .0.  /\  ( (coe1 `  F
) `  ( ( deg1  `  R ) `  F
) )  =  ( 1r `  R ) ) )
87simp2bi 971 1  |-  ( F  e.  M  ->  F  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337   Basecbs 13245   0gc0g 13499   1rcur 15438  Poly1cpl1 16351  coe1cco1 16354   deg1 cdg1 19544  Monic1pcmn1 19615
This theorem is referenced by:  mon1puc1p  19640  deg1submon1p  19642  mon1psubm  26848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-slot 13249  df-base 13250  df-mon1 19620
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