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Theorem uc1pn0 19928
Description: Unitic 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 )
uc1pn0.c  |-  C  =  (Unic1p `  R )
Assertion
Ref Expression
uc1pn0  |-  ( F  e.  C  ->  F  =/=  .0.  )

Proof of Theorem uc1pn0
StepHypRef Expression
1 uc1pn0.p . . 3  |-  P  =  (Poly1 `  R )
2 eqid 2380 . . 3  |-  ( Base `  P )  =  (
Base `  P )
3 uc1pn0.z . . 3  |-  .0.  =  ( 0g `  P )
4 eqid 2380 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 uc1pn0.c . . 3  |-  C  =  (Unic1p `  R )
6 eqid 2380 . . 3  |-  (Unit `  R )  =  (Unit `  R )
71, 2, 3, 4, 5, 6isuc1p 19923 . 2  |-  ( F  e.  C  <->  ( F  e.  ( Base `  P
)  /\  F  =/=  .0.  /\  ( (coe1 `  F
) `  ( ( deg1  `  R ) `  F
) )  e.  (Unit `  R ) ) )
87simp2bi 973 1  |-  ( F  e.  C  ->  F  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2543   ` cfv 5387   Basecbs 13389   0gc0g 13643  Unitcui 15664  Poly1cpl1 16491  coe1cco1 16494   deg1 cdg1 19837  Unic1pcuc1p 19909
This theorem is referenced by:  uc1pdeg  19930  q1peqb  19937
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-slot 13393  df-base 13394  df-uc1p 19914
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