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Theorem uc1pldg 21738
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 2451 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2451 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2451 . . 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 21730 . 2  |-  ( F  e.  C  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) )
87simp3bi 1005 1  |-  ( F  e.  C  ->  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5518   Basecbs 14278   0gc0g 14482  Unitcui 16839  Poly1cpl1 17742  coe1cco1 17743   deg1 cdg1 21641  Unic1pcuc1p 21716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fv 5526  df-slot 14282  df-base 14283  df-uc1p 21721
This theorem is referenced by:  uc1pmon1p  21741  q1peqb  21744  fta1glem1  21755  ig1peu  21761
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