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Definition df-uc1p 19517
Description: Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 19523. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Assertion
Ref Expression
df-uc1p  |- Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
Distinct variable group:    f, r

Detailed syntax breakdown of Definition df-uc1p
StepHypRef Expression
1 cuc1p 19512 . 2  class Unic1p
2 vr . . 3  set  r
3 cvv 2788 . . 3  class  _V
4 vf . . . . . . 7  set  f
54cv 1622 . . . . . 6  class  f
62cv 1622 . . . . . . . 8  class  r
7 cpl1 16252 . . . . . . . 8  class Poly1
86, 7cfv 5255 . . . . . . 7  class  (Poly1 `  r
)
9 c0g 13400 . . . . . . 7  class  0g
108, 9cfv 5255 . . . . . 6  class  ( 0g
`  (Poly1 `  r ) )
115, 10wne 2446 . . . . 5  wff  f  =/=  ( 0g `  (Poly1 `  r ) )
12 cdg1 19440 . . . . . . . . 9  class deg1
136, 12cfv 5255 . . . . . . . 8  class  ( deg1  `  r
)
145, 13cfv 5255 . . . . . . 7  class  ( ( deg1  `  r ) `  f
)
15 cco1 16255 . . . . . . . 8  class coe1
165, 15cfv 5255 . . . . . . 7  class  (coe1 `  f
)
1714, 16cfv 5255 . . . . . 6  class  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )
18 cui 15421 . . . . . . 7  class Unit
196, 18cfv 5255 . . . . . 6  class  (Unit `  r )
2017, 19wcel 1684 . . . . 5  wff  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )  e.  (Unit `  r )
2111, 20wa 358 . . . 4  wff  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )
)
22 cbs 13148 . . . . 5  class  Base
238, 22cfv 5255 . . . 4  class  ( Base `  (Poly1 `  r ) )
2421, 4, 23crab 2547 . . 3  class  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) }
252, 3, 24cmpt 4077 . 2  class  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
261, 25wceq 1623 1  wff Unic1p  =  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
Colors of variables: wff set class
This definition is referenced by:  uc1pval  19525
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