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Definition df-mhp 16103
Description: Define the subspaces of order-  n homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
df-mhp  |- mHomP  =  ( i  e.  _V , 
r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  (
i mPoly  r ) )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
)  =  n } } ) )
Distinct variable group:    f, g, h, i, j, n, r

Detailed syntax breakdown of Definition df-mhp
StepHypRef Expression
1 cmhp 16092 . 2  class mHomP
2 vi . . 3  set  i
3 vr . . 3  set  r
4 cvv 2788 . . 3  class  _V
5 vn . . . 4  set  n
6 cn0 9965 . . . 4  class  NN0
7 vf . . . . . . . . 9  set  f
87cv 1622 . . . . . . . 8  class  f
98ccnv 4688 . . . . . . 7  class  `' f
103cv 1622 . . . . . . . . . 10  class  r
11 c0g 13400 . . . . . . . . . 10  class  0g
1210, 11cfv 5255 . . . . . . . . 9  class  ( 0g
`  r )
1312csn 3640 . . . . . . . 8  class  { ( 0g `  r ) }
144, 13cdif 3149 . . . . . . 7  class  ( _V 
\  { ( 0g
`  r ) } )
159, 14cima 4692 . . . . . 6  class  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )
16 vj . . . . . . . . . . 11  set  j
1716cv 1622 . . . . . . . . . 10  class  j
18 vg . . . . . . . . . . 11  set  g
1918cv 1622 . . . . . . . . . 10  class  g
2017, 19cfv 5255 . . . . . . . . 9  class  ( g `
 j )
216, 20, 16csu 12158 . . . . . . . 8  class  sum_ j  e.  NN0  ( g `  j )
225cv 1622 . . . . . . . 8  class  n
2321, 22wceq 1623 . . . . . . 7  wff  sum_ j  e.  NN0  ( g `  j )  =  n
24 vh . . . . . . . . . . . 12  set  h
2524cv 1622 . . . . . . . . . . 11  class  h
2625ccnv 4688 . . . . . . . . . 10  class  `' h
27 cn 9746 . . . . . . . . . 10  class  NN
2826, 27cima 4692 . . . . . . . . 9  class  ( `' h " NN )
29 cfn 6863 . . . . . . . . 9  class  Fin
3028, 29wcel 1684 . . . . . . . 8  wff  ( `' h " NN )  e.  Fin
312cv 1622 . . . . . . . . 9  class  i
32 cmap 6772 . . . . . . . . 9  class  ^m
336, 31, 32co 5858 . . . . . . . 8  class  ( NN0 
^m  i )
3430, 24, 33crab 2547 . . . . . . 7  class  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }
3523, 18, 34crab 2547 . . . . . 6  class  { g  e.  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  (
g `  j )  =  n }
3615, 35wss 3152 . . . . 5  wff  ( `' f " ( _V 
\  { ( 0g
`  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  (
g `  j )  =  n }
37 cmpl 16089 . . . . . . 7  class mPoly
3831, 10, 37co 5858 . . . . . 6  class  ( i mPoly 
r )
39 cbs 13148 . . . . . 6  class  Base
4038, 39cfv 5255 . . . . 5  class  ( Base `  ( i mPoly  r ) )
4136, 7, 40crab 2547 . . . 4  class  { f  e.  ( Base `  (
i mPoly  r ) )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
)  =  n } }
425, 6, 41cmpt 4077 . . 3  class  ( n  e.  NN0  |->  { f  e.  ( Base `  (
i mPoly  r ) )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
)  =  n } } )
432, 3, 4, 4, 42cmpt2 5860 . 2  class  ( i  e.  _V ,  r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  (
i mPoly  r ) )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
)  =  n } } ) )
441, 43wceq 1623 1  wff mHomP  =  ( i  e.  _V , 
r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  (
i mPoly  r ) )  |  ( `' f
" ( _V  \  { ( 0g `  r ) } ) )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
)  =  n } } ) )
Colors of variables: wff set class
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