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Theorem mon1pldg 19535
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 2283 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2283 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2283 . . 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 19528 . 2  |-  ( F  e.  M  <->  ( F  e.  ( Base `  (Poly1 `  R ) )  /\  F  =/=  ( 0g `  (Poly1 `  R ) )  /\  ( (coe1 `  F ) `  ( D `  F ) )  =  .1.  )
)
87simp3bi 972 1  |-  ( F  e.  M  ->  (
(coe1 `  F ) `  ( D `  F ) )  =  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255   Basecbs 13148   0gc0g 13400   1rcur 15339  Poly1cpl1 16252  coe1cco1 16255   deg1 cdg1 19440  Monic1pcmn1 19511
This theorem is referenced by:  mon1puc1p  19536  deg1submon1p  19538  mon1psubm  27525
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-mon1 19516
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