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Theorem mon1pldg 20074
Description: Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
mon1pldg.d  |-  D  =  ( deg1  `  R )
mon1pldg.o  |-  .1.  =  ( 1r `  R )
mon1pldg.m  |-  M  =  (Monic1p `  R )
Assertion
Ref Expression
mon1pldg  |-  ( F  e.  M  ->  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )

Proof of Theorem mon1pldg
StepHypRef Expression
1 eqid 2438 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2438 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2438 . . 3  |-  ( 0g
`  (Poly1 `  R ) )  =  ( 0g `  (Poly1 `  R ) )
4 mon1pldg.d . . 3  |-  D  =  ( deg1  `  R )
5 mon1pldg.m . . 3  |-  M  =  (Monic1p `  R )
6 mon1pldg.o . . 3  |-  .1.  =  ( 1r `  R )
71, 2, 3, 4, 5, 6ismon1p 20067 . 2  |-  ( F  e.  M  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
)
87simp3bi 975 1  |-  ( F  e.  M  ->  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456   Basecbs 13471   0gc0g 13725   1rcur 15664  Poly1cpl1 16573  coe1cco1 16576   deg1 cdg1 19979  Monic1pcmn1 20050
This theorem is referenced by:  mon1puc1p  20075  deg1submon1p  20077  mon1psubm  27504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-slot 13475  df-base 13476  df-mon1 20055
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