MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mon1pcl Structured version   Unicode version

Theorem mon1pcl 22277
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 2467 . . 3  |-  ( 0g
`  P )  =  ( 0g `  P
)
4 eqid 2467 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
5 mon1pcl.m . . 3  |-  M  =  (Monic1p `  R )
6 eqid 2467 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
71, 2, 3, 4, 5, 6ismon1p 22275 . 2  |-  ( F  e.  M  <->  ( F  e.  B  /\  F  =/=  ( 0g `  P
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  =  ( 1r `  R
) ) )
87simp1bi 1011 1  |-  ( F  e.  M  ->  F  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5586   Basecbs 14483   0gc0g 14688   1rcur 16940  Poly1cpl1 17984  coe1cco1 17985   deg1 cdg1 22184  Monic1pcmn1 22258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-slot 14487  df-base 14488  df-mon1 22263
This theorem is referenced by:  mon1puc1p  22283  deg1submon1p  22285  ply1rem  22296  fta1glem1  22298  fta1glem2  22299  mon1psubm  30771
  Copyright terms: Public domain W3C validator