Step |
Hyp |
Ref |
Expression |
1 |
|
dia1dim2.h |
|- H = ( LHyp ` K ) |
2 |
|
dia1dim2.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
dia1dim2.r |
|- R = ( ( trL ` K ) ` W ) |
4 |
|
dva1dim2.u |
|- U = ( ( DVecA ` K ) ` W ) |
5 |
|
dia1dim2.i |
|- I = ( ( DIsoA ` K ) ` W ) |
6 |
|
dva1dim2.n |
|- N = ( LSpan ` U ) |
7 |
|
eqid |
|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W ) |
8 |
|
eqid |
|- ( Scalar ` U ) = ( Scalar ` U ) |
9 |
|
eqid |
|- ( Base ` ( Scalar ` U ) ) = ( Base ` ( Scalar ` U ) ) |
10 |
1 7 4 8 9
|
dvabase |
|- ( ( K e. HL /\ W e. H ) -> ( Base ` ( Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) ) |
11 |
10
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( Base ` ( Scalar ` U ) ) = ( ( TEndo ` K ) ` W ) ) |
12 |
11
|
rexeqdv |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( E. s e. ( Base ` ( Scalar ` U ) ) g = ( s ( .s ` U ) F ) <-> E. s e. ( ( TEndo ` K ) ` W ) g = ( s ( .s ` U ) F ) ) ) |
13 |
|
eqid |
|- ( .s ` U ) = ( .s ` U ) |
14 |
1 2 7 4 13
|
dvavsca |
|- ( ( ( K e. HL /\ W e. H ) /\ ( s e. ( ( TEndo ` K ) ` W ) /\ F e. T ) ) -> ( s ( .s ` U ) F ) = ( s ` F ) ) |
15 |
14
|
anass1rs |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ s e. ( ( TEndo ` K ) ` W ) ) -> ( s ( .s ` U ) F ) = ( s ` F ) ) |
16 |
15
|
eqeq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T ) /\ s e. ( ( TEndo ` K ) ` W ) ) -> ( g = ( s ( .s ` U ) F ) <-> g = ( s ` F ) ) ) |
17 |
16
|
rexbidva |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( E. s e. ( ( TEndo ` K ) ` W ) g = ( s ( .s ` U ) F ) <-> E. s e. ( ( TEndo ` K ) ` W ) g = ( s ` F ) ) ) |
18 |
12 17
|
bitrd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( E. s e. ( Base ` ( Scalar ` U ) ) g = ( s ( .s ` U ) F ) <-> E. s e. ( ( TEndo ` K ) ` W ) g = ( s ` F ) ) ) |
19 |
18
|
abbidv |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> { g | E. s e. ( Base ` ( Scalar ` U ) ) g = ( s ( .s ` U ) F ) } = { g | E. s e. ( ( TEndo ` K ) ` W ) g = ( s ` F ) } ) |
20 |
1 4
|
dvalvec |
|- ( ( K e. HL /\ W e. H ) -> U e. LVec ) |
21 |
20
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> U e. LVec ) |
22 |
|
lveclmod |
|- ( U e. LVec -> U e. LMod ) |
23 |
21 22
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> U e. LMod ) |
24 |
|
simpr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F e. T ) |
25 |
|
eqid |
|- ( Base ` U ) = ( Base ` U ) |
26 |
1 2 4 25
|
dvavbase |
|- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = T ) |
27 |
26
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( Base ` U ) = T ) |
28 |
24 27
|
eleqtrrd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F e. ( Base ` U ) ) |
29 |
8 9 25 13 6
|
lspsn |
|- ( ( U e. LMod /\ F e. ( Base ` U ) ) -> ( N ` { F } ) = { g | E. s e. ( Base ` ( Scalar ` U ) ) g = ( s ( .s ` U ) F ) } ) |
30 |
23 28 29
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( N ` { F } ) = { g | E. s e. ( Base ` ( Scalar ` U ) ) g = ( s ( .s ` U ) F ) } ) |
31 |
1 2 3 7 5
|
dia1dim |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = { g | E. s e. ( ( TEndo ` K ) ` W ) g = ( s ` F ) } ) |
32 |
19 30 31
|
3eqtr4rd |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = ( N ` { F } ) ) |