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Definition df-uc1p 19923
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 19929. (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 19918 . 2  class Unic1p
2 vr . . 3  set  r
3 cvv 2901 . . 3  class  _V
4 vf . . . . . . 7  set  f
54cv 1648 . . . . . 6  class  f
62cv 1648 . . . . . . . 8  class  r
7 cpl1 16500 . . . . . . . 8  class Poly1
86, 7cfv 5396 . . . . . . 7  class  (Poly1 `  r
)
9 c0g 13652 . . . . . . 7  class  0g
108, 9cfv 5396 . . . . . 6  class  ( 0g
`  (Poly1 `  r ) )
115, 10wne 2552 . . . . 5  wff  f  =/=  ( 0g `  (Poly1 `  r ) )
12 cdg1 19846 . . . . . . . . 9  class deg1
136, 12cfv 5396 . . . . . . . 8  class  ( deg1  `  r
)
145, 13cfv 5396 . . . . . . 7  class  ( ( deg1  `  r ) `  f
)
15 cco1 16503 . . . . . . . 8  class coe1
165, 15cfv 5396 . . . . . . 7  class  (coe1 `  f
)
1714, 16cfv 5396 . . . . . 6  class  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )
18 cui 15673 . . . . . . 7  class Unit
196, 18cfv 5396 . . . . . 6  class  (Unit `  r )
2017, 19wcel 1717 . . . . 5  wff  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )  e.  (Unit `  r )
2111, 20wa 359 . . . 4  wff  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )
)
22 cbs 13398 . . . . 5  class  Base
238, 22cfv 5396 . . . 4  class  ( Base `  (Poly1 `  r ) )
2421, 4, 23crab 2655 . . 3  class  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) }
252, 3, 24cmpt 4209 . 2  class  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
261, 25wceq 1649 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  19931
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