Step |
Hyp |
Ref |
Expression |
1 |
|
dicvaddcl.l |
|- .<_ = ( le ` K ) |
2 |
|
dicvaddcl.a |
|- A = ( Atoms ` K ) |
3 |
|
dicvaddcl.h |
|- H = ( LHyp ` K ) |
4 |
|
dicvaddcl.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
dicvaddcl.i |
|- I = ( ( DIsoC ` K ) ` W ) |
6 |
|
dicvaddcl.p |
|- .+ = ( +g ` U ) |
7 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( K e. HL /\ W e. H ) ) |
8 |
|
eqid |
|- ( Base ` U ) = ( Base ` U ) |
9 |
1 2 3 5 4 8
|
dicssdvh |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` U ) ) |
10 |
|
eqid |
|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W ) |
11 |
|
eqid |
|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W ) |
12 |
3 10 11 4 8
|
dvhvbase |
|- ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) |
13 |
12
|
eqcomd |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) ) |
14 |
13
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) = ( Base ` U ) ) |
15 |
9 14
|
sseqtrrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) |
16 |
15
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) |
17 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> X e. ( I ` Q ) ) |
18 |
16 17
|
sseldd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> X e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) |
19 |
|
simp3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> Y e. ( I ` Q ) ) |
20 |
16 19
|
sseldd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> Y e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) |
21 |
|
eqid |
|- ( Scalar ` U ) = ( Scalar ` U ) |
22 |
|
eqid |
|- ( +g ` ( Scalar ` U ) ) = ( +g ` ( Scalar ` U ) ) |
23 |
3 10 11 4 21 6 22
|
dvhvadd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) /\ Y e. ( ( ( LTrn ` K ) ` W ) X. ( ( TEndo ` K ) ` W ) ) ) ) -> ( X .+ Y ) = <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) >. ) |
24 |
7 18 20 23
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) = <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) >. ) |
25 |
1 2 3 11 5
|
dicelval2nd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. ( I ` Q ) ) -> ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) ) |
26 |
25
|
3adant3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) ) |
27 |
1 2 3 11 5
|
dicelval2nd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) |
28 |
27
|
3adant3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) |
29 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
30 |
1 29 2 3
|
lhpocnel |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) ) |
31 |
30
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) ) |
32 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
33 |
|
eqid |
|- ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) |
34 |
1 2 3 10 33
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) ) |
35 |
7 31 32 34
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) ) |
36 |
|
eqid |
|- ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) |
37 |
10 36
|
tendospdi2 |
|- ( ( ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) /\ ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) e. ( ( LTrn ` K ) ` W ) ) -> ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) ) |
38 |
26 28 35 37
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) ) |
39 |
3 10 11 4 21 36 22
|
dvhfplusr |
|- ( ( K e. HL /\ W e. H ) -> ( +g ` ( Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ) |
40 |
39
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( +g ` ( Scalar ` U ) ) = ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ) |
41 |
40
|
oveqd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) = ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ) |
42 |
41
|
fveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) = ( ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
43 |
|
eqid |
|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W ) |
44 |
1 2 3 43 10 5
|
dicelval1sta |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. ( I ` Q ) ) -> ( 1st ` X ) = ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
45 |
44
|
3adant3r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 1st ` X ) = ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
46 |
1 2 3 43 10 5
|
dicelval1sta |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
47 |
46
|
3adant3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
48 |
45 47
|
coeq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) o. ( ( 2nd ` Y ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) ) |
49 |
38 42 48
|
3eqtr4rd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) ) |
50 |
3 10 11 36
|
tendoplcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( 2nd ` X ) e. ( ( TEndo ` K ) ` W ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) -> ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) |
51 |
7 26 28 50
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( s e. ( ( TEndo ` K ) ` W ) , t e. ( ( TEndo ` K ) ` W ) |-> ( h e. ( ( LTrn ` K ) ` W ) |-> ( ( s ` h ) o. ( t ` h ) ) ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) |
52 |
41 51
|
eqeltrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) |
53 |
|
fvex |
|- ( 1st ` X ) e. _V |
54 |
|
fvex |
|- ( 1st ` Y ) e. _V |
55 |
53 54
|
coex |
|- ( ( 1st ` X ) o. ( 1st ` Y ) ) e. _V |
56 |
|
ovex |
|- ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) e. _V |
57 |
1 2 3 43 10 11 5 55 56
|
dicopelval |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) <-> ( ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) ) ) |
58 |
57
|
3adant3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) <-> ( ( ( 1st ` X ) o. ( 1st ` Y ) ) = ( ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) ) /\ ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) e. ( ( TEndo ` K ) ` W ) ) ) ) |
59 |
49 52 58
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> <. ( ( 1st ` X ) o. ( 1st ` Y ) ) , ( ( 2nd ` X ) ( +g ` ( Scalar ` U ) ) ( 2nd ` Y ) ) >. e. ( I ` Q ) ) |
60 |
24 59
|
eqeltrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) e. ( I ` Q ) ) |