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Theorem uc1pcl 22272
Description: Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pcl.p  |-  P  =  (Poly1 `  R )
uc1pcl.b  |-  B  =  ( Base `  P
)
uc1pcl.c  |-  C  =  (Unic1p `  R )
Assertion
Ref Expression
uc1pcl  |-  ( F  e.  C  ->  F  e.  B )

Proof of Theorem uc1pcl
StepHypRef Expression
1 uc1pcl.p . . 3  |-  P  =  (Poly1 `  R )
2 uc1pcl.b . . 3  |-  B  =  ( Base `  P
)
3 eqid 2460 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2460 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 uc1pcl.c . . 3  |-  C  =  (Unic1p `  R )
6 eqid 2460 . . 3  |-  (Unit `  R )  =  (Unit `  R )
71, 2, 3, 4, 5, 6isuc1p 22269 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  e.  (Unit `  R )
) )
87simp1bi 1006 1  |-  ( F  e.  C  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    =/= wne 2655   ` cfv 5579   Basecbs 14479   0gc0g 14684  Unitcui 17065  Poly1cpl1 17980  coe1cco1 17981   deg1 cdg1 22180  Unic1pcuc1p 22255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-slot 14483  df-base 14484  df-uc1p 22260
This theorem is referenced by:  uc1pdeg  22276  uc1pmon1p  22280  q1peqb  22283  r1pcl  22286  r1pdeglt  22287  r1pid  22288  dvdsq1p  22289  dvdsr1p  22290
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