Theorem hdmap1ffval

Description: Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015)

	Ref	Expression
Hypothesis	hdmap1val.h	|- H = ( LHyp ` K )
Assertion	hdmap1ffval	|- ( K e. X -> ( HDMap1 ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )

Proof

Step	Hyp	Ref	Expression
1		hdmap1val.h	|- H = ( LHyp ` K )
2		elex	|- ( K e. X -> K e. _V )
3		fveq2	|- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
4	3 1	eqtr4di	|- ( k = K -> ( LHyp ` k ) = H )
5		fveq2	|- ( k = K -> ( DVecH ` k ) = ( DVecH ` K ) )
6	5	fveq1d	|- ( k = K -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` w ) )
7		fveq2	|- ( k = K -> ( LCDual ` k ) = ( LCDual ` K ) )
8	7	fveq1d	|- ( k = K -> ( ( LCDual ` k ) ` w ) = ( ( LCDual ` K ) ` w ) )
9		fveq2	|- ( k = K -> ( mapd ` k ) = ( mapd ` K ) )
10	9	fveq1d	|- ( k = K -> ( ( mapd ` k ) ` w ) = ( ( mapd ` K ) ` w ) )
11	10	sbceq1d	|- ( k = K -> ( [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
12	11	sbcbidv	|- ( k = K -> ( [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
13	12	sbcbidv	|- ( k = K -> ( [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
14	8 13	sbceqbid	|- ( k = K -> ( [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
15	14	sbcbidv	|- ( k = K -> ( [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
16	15	sbcbidv	|- ( k = K -> ( [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
17	6 16	sbceqbid	|- ( k = K -> ( [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) <-> [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) ) )
18	17	abbidv	|- ( k = K -> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } = { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } )
19	4 18	mpteq12dv	|- ( k = K -> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )
20		df-hdmap1	|- HDMap1 = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> { a | [. ( ( DVecH ` k ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` k ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` k ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )
21	19 20 1	mptfvmpt	|- ( K e. _V -> ( HDMap1 ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )
22	2 21	syl	|- ( K e. X -> ( HDMap1 ` K ) = ( w e. H |-> { a | [. ( ( DVecH ` K ) ` w ) / u ]. [. ( Base ` u ) / v ]. [. ( LSpan ` u ) / n ]. [. ( ( LCDual ` K ) ` w ) / c ]. [. ( Base ` c ) / d ]. [. ( LSpan ` c ) / j ]. [. ( ( mapd ` K ) ` w ) / m ]. a e. ( x e. ( ( v X. d ) X. v ) |-> if ( ( 2nd ` x ) = ( 0g ` u ) , ( 0g ` c ) , ( iota_ h e. d ( ( m ` ( n ` { ( 2nd ` x ) } ) ) = ( j ` { h } ) /\ ( m ` ( n ` { ( ( 1st ` ( 1st ` x ) ) ( -g ` u ) ( 2nd ` x ) ) } ) ) = ( j ` { ( ( 2nd ` ( 1st ` x ) ) ( -g ` c ) h ) } ) ) ) ) ) } ) )