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Theorem uc1pldg 19749
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 2366 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2366 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2366 . . 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 19741 . 2  |-  ( F  e.  C  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  e.  U ) )
87simp3bi 973 1  |-  ( F  e.  C  ->  (
(coe1 `  F ) `  ( D `  F ) )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715    =/= wne 2529   ` cfv 5358   Basecbs 13356   0gc0g 13610  Unitcui 15631  Poly1cpl1 16462  coe1cco1 16465   deg1 cdg1 19655  Unic1pcuc1p 19727
This theorem is referenced by:  uc1pmon1p  19752  q1peqb  19755  fta1glem1  19766  ig1peu  19772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-slot 13360  df-base 13361  df-uc1p 19732
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