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Definition df-docaN 31932
Description: Define subspace orthocomplement for  DVecA partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)
Assertion
Ref Expression
df-docaN  |-  ocA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P (
( LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) ) ) )
Distinct variable group:    w, k, x, z

Detailed syntax breakdown of Definition df-docaN
StepHypRef Expression
1 cocaN 31931 . 2  class  ocA
2 vk . . 3  set  k
3 cvv 2801 . . 3  class  _V
4 vw . . . 4  set  w
52cv 1631 . . . . 5  class  k
6 clh 30795 . . . . 5  class  LHyp
75, 6cfv 5271 . . . 4  class  ( LHyp `  k )
8 vx . . . . 5  set  x
94cv 1631 . . . . . . 7  class  w
10 cltrn 30912 . . . . . . . 8  class  LTrn
115, 10cfv 5271 . . . . . . 7  class  ( LTrn `  k )
129, 11cfv 5271 . . . . . 6  class  ( (
LTrn `  k ) `  w )
1312cpw 3638 . . . . 5  class  ~P (
( LTrn `  k ) `  w )
148cv 1631 . . . . . . . . . . . . 13  class  x
15 vz . . . . . . . . . . . . . 14  set  z
1615cv 1631 . . . . . . . . . . . . 13  class  z
1714, 16wss 3165 . . . . . . . . . . . 12  wff  x  C_  z
18 cdia 31840 . . . . . . . . . . . . . . 15  class  DIsoA
195, 18cfv 5271 . . . . . . . . . . . . . 14  class  ( DIsoA `  k )
209, 19cfv 5271 . . . . . . . . . . . . 13  class  ( (
DIsoA `  k ) `  w )
2120crn 4706 . . . . . . . . . . . 12  class  ran  (
( DIsoA `  k ) `  w )
2217, 15, 21crab 2560 . . . . . . . . . . 11  class  { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }
2322cint 3878 . . . . . . . . . 10  class  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z }
2420ccnv 4704 . . . . . . . . . 10  class  `' ( ( DIsoA `  k ) `  w )
2523, 24cfv 5271 . . . . . . . . 9  class  ( `' ( ( DIsoA `  k
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  k ) `  w
)  |  x  C_  z } )
26 coc 13232 . . . . . . . . . 10  class  oc
275, 26cfv 5271 . . . . . . . . 9  class  ( oc
`  k )
2825, 27cfv 5271 . . . . . . . 8  class  ( ( oc `  k ) `
 ( `' ( ( DIsoA `  k ) `  w ) `  |^| { z  e.  ran  (
( DIsoA `  k ) `  w )  |  x 
C_  z } ) )
299, 27cfv 5271 . . . . . . . 8  class  ( ( oc `  k ) `
 w )
30 cjn 14094 . . . . . . . . 9  class  join
315, 30cfv 5271 . . . . . . . 8  class  ( join `  k )
3228, 29, 31co 5874 . . . . . . 7  class  ( ( ( oc `  k
) `  ( `' ( ( DIsoA `  k
) `  w ) `  |^| { z  e. 
ran  ( ( DIsoA `  k ) `  w
)  |  x  C_  z } ) ) (
join `  k )
( ( oc `  k ) `  w
) )
33 cmee 14095 . . . . . . . 8  class  meet
345, 33cfv 5271 . . . . . . 7  class  ( meet `  k )
3532, 9, 34co 5874 . . . . . 6  class  ( ( ( ( oc `  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w )
3635, 20cfv 5271 . . . . 5  class  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) )
378, 13, 36cmpt 4093 . . . 4  class  ( x  e.  ~P ( (
LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) )
384, 7, 37cmpt 4093 . . 3  class  ( w  e.  ( LHyp `  k
)  |->  ( x  e. 
~P ( ( LTrn `  k ) `  w
)  |->  ( ( (
DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) ) )
392, 3, 38cmpt 4093 . 2  class  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k
)  |->  ( x  e. 
~P ( ( LTrn `  k ) `  w
)  |->  ( ( (
DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) ) ) )
401, 39wceq 1632 1  wff  ocA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( x  e.  ~P (
( LTrn `  k ) `  w )  |->  ( ( ( DIsoA `  k ) `  w ) `  (
( ( ( oc
`  k ) `  ( `' ( ( DIsoA `  k ) `  w
) `  |^| { z  e.  ran  ( (
DIsoA `  k ) `  w )  |  x 
C_  z } ) ) ( join `  k
) ( ( oc
`  k ) `  w ) ) (
meet `  k )
w ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  docaffvalN  31933
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