Metamath Proof Explorer

Theorem hdmap1vallem

Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 15-May-2015)

Ref Expression
Hypotheses hdmap1val.h 
|- H = ( LHyp ` K )
hdmap1fval.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1fval.v 
|- V = ( Base ` U )
hdmap1fval.s 
|- .- = ( -g ` U )
hdmap1fval.o 
|- .0. = ( 0g ` U )
hdmap1fval.n 
|- N = ( LSpan ` U )
hdmap1fval.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1fval.d 
|- D = ( Base ` C )
hdmap1fval.r 
|- R = ( -g ` C )
hdmap1fval.q 
|- Q = ( 0g ` C )
hdmap1fval.j 
|- J = ( LSpan ` C )
hdmap1fval.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1fval.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1fval.k 
|- ( ph -> ( K e. A /\ W e. H ) )
hdmap1val.t 
|- ( ph -> T e. ( ( V X. D ) X. V ) )
Assertion hdmap1vallem 
|- ( ph -> ( I ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1val.h 
 |-  H = ( LHyp ` K )
2 hdmap1fval.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1fval.v 
 |-  V = ( Base ` U )
4 hdmap1fval.s 
 |-  .- = ( -g ` U )
5 hdmap1fval.o 
 |-  .0. = ( 0g ` U )
6 hdmap1fval.n 
 |-  N = ( LSpan ` U )
7 hdmap1fval.c 
 |-  C = ( ( LCDual ` K ) ` W )
8 hdmap1fval.d 
 |-  D = ( Base ` C )
9 hdmap1fval.r 
 |-  R = ( -g ` C )
10 hdmap1fval.q 
 |-  Q = ( 0g ` C )
11 hdmap1fval.j 
 |-  J = ( LSpan ` C )
12 hdmap1fval.m 
 |-  M = ( ( mapd ` K ) ` W )
13 hdmap1fval.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
14 hdmap1fval.k 
 |-  ( ph -> ( K e. A /\ W e. H ) )
15 hdmap1val.t 
 |-  ( ph -> T e. ( ( V X. D ) X. V ) )
16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hdmap1fval 
 |-  ( ph -> I = ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) )
17 16 fveq1d 
 |-  ( ph -> ( I ` T ) = ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) )
18 10 fvexi 
 |-  Q e. _V
19 riotaex 
 |-  ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) e. _V
20 18 19 ifex 
 |-  if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) e. _V
21 fveqeq2 
 |-  ( x = T -> ( ( 2nd ` x ) = .0. <-> ( 2nd ` T ) = .0. ) )
22 fveq2 
 |-  ( x = T -> ( 2nd ` x ) = ( 2nd ` T ) )
23 22 sneqd 
 |-  ( x = T -> { ( 2nd ` x ) } = { ( 2nd ` T ) } )
24 23 fveq2d 
 |-  ( x = T -> ( N ` { ( 2nd ` x ) } ) = ( N ` { ( 2nd ` T ) } ) )
25 24 fveqeq2d 
 |-  ( x = T -> ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) ) )
26 2fveq3 
 |-  ( x = T -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` T ) ) )
27 26 22 oveq12d 
 |-  ( x = T -> ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) = ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) )
28 27 sneqd 
 |-  ( x = T -> { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } = { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } )
29 28 fveq2d 
 |-  ( x = T -> ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) = ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) )
30 29 fveq2d 
 |-  ( x = T -> ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) )
31 2fveq3 
 |-  ( x = T -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` T ) ) )
32 31 oveq1d 
 |-  ( x = T -> ( ( 2nd ` ( 1st ` x ) ) R h ) = ( ( 2nd ` ( 1st ` T ) ) R h ) )
33 32 sneqd 
 |-  ( x = T -> { ( ( 2nd ` ( 1st ` x ) ) R h ) } = { ( ( 2nd ` ( 1st ` T ) ) R h ) } )
34 33 fveq2d 
 |-  ( x = T -> ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) )
35 30 34 eqeq12d 
 |-  ( x = T -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) <-> ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) )
36 25 35 anbi12d 
 |-  ( x = T -> ( ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) <-> ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) )
37 36 riotabidv 
 |-  ( x = T -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) )
38 21 37 ifbieq2d 
 |-  ( x = T -> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
39 eqid 
 |-  ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) = ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
40 38 39 fvmptg 
 |-  ( ( T e. ( ( V X. D ) X. V ) /\ if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) e. _V ) -> ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
41 15 20 40 sylancl 
 |-  ( ph -> ( ( x e. ( ( V X. D ) X. V ) |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )
42 17 41 eqtrd 
 |-  ( ph -> ( I ` T ) = if ( ( 2nd ` T ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` T ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` T ) ) .- ( 2nd ` T ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` T ) ) R h ) } ) ) ) ) )