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Definition df-uc1p 21618
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 21624. (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 21613 . 2  class Unic1p
2 vr . . 3  setvar  r
3 cvv 2987 . . 3  class  _V
4 vf . . . . . . 7  setvar  f
54cv 1368 . . . . . 6  class  f
62cv 1368 . . . . . . . 8  class  r
7 cpl1 17648 . . . . . . . 8  class Poly1
86, 7cfv 5433 . . . . . . 7  class  (Poly1 `  r
)
9 c0g 14393 . . . . . . 7  class  0g
108, 9cfv 5433 . . . . . 6  class  ( 0g
`  (Poly1 `  r ) )
115, 10wne 2620 . . . . 5  wff  f  =/=  ( 0g `  (Poly1 `  r ) )
12 cdg1 21538 . . . . . . . . 9  class deg1
136, 12cfv 5433 . . . . . . . 8  class  ( deg1  `  r
)
145, 13cfv 5433 . . . . . . 7  class  ( ( deg1  `  r ) `  f
)
15 cco1 17649 . . . . . . . 8  class coe1
165, 15cfv 5433 . . . . . . 7  class  (coe1 `  f
)
1714, 16cfv 5433 . . . . . 6  class  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )
18 cui 16746 . . . . . . 7  class Unit
196, 18cfv 5433 . . . . . 6  class  (Unit `  r )
2017, 19wcel 1756 . . . . 5  wff  ( (coe1 `  f ) `  (
( deg1  `
 r ) `  f ) )  e.  (Unit `  r )
2111, 20wa 369 . . . 4  wff  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f ) `  ( ( deg1  `  r ) `  f ) )  e.  (Unit `  r )
)
22 cbs 14189 . . . . 5  class  Base
238, 22cfv 5433 . . . 4  class  ( Base `  (Poly1 `  r ) )
2421, 4, 23crab 2734 . . 3  class  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) }
252, 3, 24cmpt 4365 . 2  class  ( r  e.  _V  |->  { f  e.  ( Base `  (Poly1 `  r ) )  |  ( f  =/=  ( 0g `  (Poly1 `  r ) )  /\  ( (coe1 `  f
) `  ( ( deg1  `  r ) `  f
) )  e.  (Unit `  r ) ) } )
261, 25wceq 1369 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 setvar class
This definition is referenced by:  uc1pval  21626
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