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Theorem uc1pcl 19529
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 2283 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2283 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 uc1pcl.c . . 3  |-  C  =  (Unic1p `  R )
6 eqid 2283 . . 3  |-  (Unit `  R )  =  (Unit `  R )
71, 2, 3, 4, 5, 6isuc1p 19526 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  e.  (Unit `  R )
) )
87simp1bi 970 1  |-  ( F  e.  C  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   Basecbs 13148   0gc0g 13400  Unitcui 15421  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  Unic1pcuc1p 19512
This theorem is referenced by:  uc1pdeg  19533  uc1pmon1p  19537  q1peqb  19540  r1pcl  19543  r1pdeglt  19544  r1pid  19545  dvdsq1p  19546  dvdsr1p  19547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-uc1p 19517
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