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Definition df-irng 23966
Description: Define the subring of elements of  r integral over  s in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-irng  |- IntgRing  =  ( r  e.  _V , 
s  e.  _V  |->  U_ f  e.  (Monic1p `  (
rs  s ) ) ( `' f " {
( 0g `  r
) } ) )
Distinct variable group:    f, r, s

Detailed syntax breakdown of Definition df-irng
StepHypRef Expression
1 citr 23958 . 2  class IntgRing
2 vr . . 3  set  r
3 vs . . 3  set  s
4 cvv 2788 . . 3  class  _V
5 vf . . . 4  set  f
62cv 1622 . . . . . 6  class  r
73cv 1622 . . . . . 6  class  s
8 cress 13149 . . . . . 6  classs
96, 7, 8co 5858 . . . . 5  class  ( rs  s )
10 cmn1 19511 . . . . 5  class Monic1p
119, 10cfv 5255 . . . 4  class  (Monic1p `  (
rs  s ) )
125cv 1622 . . . . . 6  class  f
1312ccnv 4688 . . . . 5  class  `' f
14 c0g 13400 . . . . . . 7  class  0g
156, 14cfv 5255 . . . . . 6  class  ( 0g
`  r )
1615csn 3640 . . . . 5  class  { ( 0g `  r ) }
1713, 16cima 4692 . . . 4  class  ( `' f " { ( 0g `  r ) } )
185, 11, 17ciun 3905 . . 3  class  U_ f  e.  (Monic1p `  ( rs  s ) ) ( `' f
" { ( 0g
`  r ) } )
192, 3, 4, 4, 18cmpt2 5860 . 2  class  ( r  e.  _V ,  s  e.  _V  |->  U_ f  e.  (Monic1p `  ( rs  s ) ) ( `' f
" { ( 0g
`  r ) } ) )
201, 19wceq 1623 1  wff IntgRing  =  ( r  e.  _V , 
s  e.  _V  |->  U_ f  e.  (Monic1p `  (
rs  s ) ) ( `' f " {
( 0g `  r
) } ) )
Colors of variables: wff set class
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