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Theorem mon1pldg 19551
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 2296 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2296 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2296 . . 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 19544 . 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 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271   Basecbs 13164   0gc0g 13416   1rcur 15355  Poly1cpl1 16268  coe1cco1 16271   deg1 cdg1 19456  Monic1pcmn1 19527
This theorem is referenced by:  mon1puc1p  19552  deg1submon1p  19554  mon1psubm  27628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168  df-base 13169  df-mon1 19532
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