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Theorem uc1pldg 20028
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 2408 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2408 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2408 . . 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 20020 . 2  |-  ( F  e.  C  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) )
87simp3bi 974 1  |-  ( F  e.  C  ->  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571   ` cfv 5417   Basecbs 13428   0gc0g 13682  Unitcui 15703  Poly1cpl1 16530  coe1cco1 16533   deg1 cdg1 19934  Unic1pcuc1p 20006
This theorem is referenced by:  uc1pmon1p  20031  q1peqb  20034  fta1glem1  20045  ig1peu  20051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-slot 13432  df-base 13433  df-uc1p 20011
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