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Theorem uc1pldg 22715
Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pldg.d  |-  D  =  ( deg1  `  R )
uc1pldg.u  |-  U  =  (Unit `  R )
uc1pldg.c  |-  C  =  (Unic1p `  R )
Assertion
Ref Expression
uc1pldg  |-  ( F  e.  C  ->  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )

Proof of Theorem uc1pldg
StepHypRef Expression
1 eqid 2454 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2454 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2454 . . 3  |-  ( 0g
`  (Poly1 `  R ) )  =  ( 0g `  (Poly1 `  R ) )
4 uc1pldg.d . . 3  |-  D  =  ( deg1  `  R )
5 uc1pldg.c . . 3  |-  C  =  (Unic1p `  R )
6 uc1pldg.u . . 3  |-  U  =  (Unit `  R )
71, 2, 3, 4, 5, 6isuc1p 22707 . 2  |-  ( F  e.  C  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) )
87simp3bi 1011 1  |-  ( F  e.  C  ->  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570   Basecbs 14716   0gc0g 14929  Unitcui 17483  Poly1cpl1 18411  coe1cco1 18412   deg1 cdg1 22618  Unic1pcuc1p 22693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-slot 14720  df-base 14721  df-uc1p 22698
This theorem is referenced by:  uc1pmon1p  22718  q1peqb  22721  fta1glem1  22732  ig1peu  22738
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