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Theorem uc1pcl 22834
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 2402 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2402 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 uc1pcl.c . . 3  |-  C  =  (Unic1p `  R )
6 eqid 2402 . . 3  |-  (Unit `  R )  =  (Unit `  R )
71, 2, 3, 4, 5, 6isuc1p 22831 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  e.  (Unit `  R )
) )
87simp1bi 1012 1  |-  ( F  e.  C  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5568   Basecbs 14839   0gc0g 15052  Unitcui 17606  Poly1cpl1 18534  coe1cco1 18535   deg1 cdg1 22742  Unic1pcuc1p 22817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-slot 14843  df-base 14844  df-uc1p 22822
This theorem is referenced by:  uc1pdeg  22838  uc1pmon1p  22842  q1peqb  22845  r1pcl  22848  r1pdeglt  22849  r1pid  22850  dvdsq1p  22851  dvdsr1p  22852
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