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Theorem mon1pcl 19935
Description: Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pcl.p  |-  P  =  (Poly1 `  R )
uc1pcl.b  |-  B  =  ( Base `  P
)
mon1pcl.m  |-  M  =  (Monic1p `  R )
Assertion
Ref Expression
mon1pcl  |-  ( F  e.  M  ->  F  e.  B )

Proof of Theorem mon1pcl
StepHypRef Expression
1 uc1pcl.p . . 3  |-  P  =  (Poly1 `  R )
2 uc1pcl.b . . 3  |-  B  =  ( Base `  P
)
3 eqid 2388 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2388 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pcl.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2388 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 19933 . 2  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  =  ( 1r `  R
) ) )
87simp1bi 972 1  |-  ( F  e.  M  ->  F  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2551   ` cfv 5395   Basecbs 13397   0gc0g 13651   1rcur 15590  Poly1cpl1 16499  coe1cco1 16502   deg1 cdg1 19845  Monic1pcmn1 19916
This theorem is referenced by:  mon1puc1p  19941  deg1submon1p  19943  ply1rem  19954  fta1glem1  19956  fta1glem2  19957  mon1psubm  27195
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fv 5403  df-slot 13401  df-base 13402  df-mon1 19921
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