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Theorem mon1pcl 21744
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 2452 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2452 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pcl.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2452 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 21742 . 2  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  =  ( 1r `  R
) ) )
87simp1bi 1003 1  |-  ( F  e.  M  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    =/= wne 2645   ` cfv 5521   Basecbs 14287   0gc0g 14492   1rcur 16720  Poly1cpl1 17752  coe1cco1 17753   deg1 cdg1 21651  Monic1pcmn1 21725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-slot 14291  df-base 14292  df-mon1 21730
This theorem is referenced by:  mon1puc1p  21750  deg1submon1p  21752  ply1rem  21763  fta1glem1  21765  fta1glem2  21766  mon1psubm  29717
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