Step |
Hyp |
Ref |
Expression |
1 |
|
dihopellsm.b |
|- B = ( Base ` K ) |
2 |
|
dihopellsm.h |
|- H = ( LHyp ` K ) |
3 |
|
dihopellsm.t |
|- T = ( ( LTrn ` K ) ` W ) |
4 |
|
dihopellsm.e |
|- E = ( ( TEndo ` K ) ` W ) |
5 |
|
dihopellsm.a |
|- A = ( v e. E , w e. E |-> ( i e. T |-> ( ( v ` i ) o. ( w ` i ) ) ) ) |
6 |
|
dihopellsm.u |
|- U = ( ( DVecH ` K ) ` W ) |
7 |
|
dihopellsm.l |
|- L = ( LSubSp ` U ) |
8 |
|
dihopellsm.p |
|- .(+) = ( LSSum ` U ) |
9 |
|
dihopellsm.i |
|- I = ( ( DIsoH ` K ) ` W ) |
10 |
|
dihopellsm.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
11 |
|
dihopellsm.x |
|- ( ph -> X e. B ) |
12 |
|
dihopellsm.y |
|- ( ph -> Y e. B ) |
13 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
14 |
1 2 9 6 13
|
dihlss |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. ( LSubSp ` U ) ) |
15 |
10 11 14
|
syl2anc |
|- ( ph -> ( I ` X ) e. ( LSubSp ` U ) ) |
16 |
1 2 9 6 13
|
dihlss |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. B ) -> ( I ` Y ) e. ( LSubSp ` U ) ) |
17 |
10 12 16
|
syl2anc |
|- ( ph -> ( I ` Y ) e. ( LSubSp ` U ) ) |
18 |
|
eqid |
|- ( +g ` U ) = ( +g ` U ) |
19 |
2 6 18 13 8
|
dvhopellsm |
|- ( ( ( K e. HL /\ W e. H ) /\ ( I ` X ) e. ( LSubSp ` U ) /\ ( I ` Y ) e. ( LSubSp ` U ) ) -> ( <. F , S >. e. ( ( I ` X ) .(+) ( I ` Y ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) ) ) ) |
20 |
10 15 17 19
|
syl3anc |
|- ( ph -> ( <. F , S >. e. ( ( I ` X ) .(+) ( I ` Y ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) ) ) ) |
21 |
10
|
adantr |
|- ( ( ph /\ <. g , t >. e. ( I ` X ) ) -> ( K e. HL /\ W e. H ) ) |
22 |
11
|
adantr |
|- ( ( ph /\ <. g , t >. e. ( I ` X ) ) -> X e. B ) |
23 |
|
simpr |
|- ( ( ph /\ <. g , t >. e. ( I ` X ) ) -> <. g , t >. e. ( I ` X ) ) |
24 |
1 2 3 4 9 21 22 23
|
dihopcl |
|- ( ( ph /\ <. g , t >. e. ( I ` X ) ) -> ( g e. T /\ t e. E ) ) |
25 |
10
|
adantr |
|- ( ( ph /\ <. h , u >. e. ( I ` Y ) ) -> ( K e. HL /\ W e. H ) ) |
26 |
12
|
adantr |
|- ( ( ph /\ <. h , u >. e. ( I ` Y ) ) -> Y e. B ) |
27 |
|
simpr |
|- ( ( ph /\ <. h , u >. e. ( I ` Y ) ) -> <. h , u >. e. ( I ` Y ) ) |
28 |
1 2 3 4 9 25 26 27
|
dihopcl |
|- ( ( ph /\ <. h , u >. e. ( I ` Y ) ) -> ( h e. T /\ u e. E ) ) |
29 |
24 28
|
anim12dan |
|- ( ( ph /\ ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) ) -> ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) |
30 |
10
|
adantr |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( K e. HL /\ W e. H ) ) |
31 |
|
simprl |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( g e. T /\ t e. E ) ) |
32 |
|
simprr |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( h e. T /\ u e. E ) ) |
33 |
2 3 4 5 6 18
|
dvhopvadd2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) -> ( <. g , t >. ( +g ` U ) <. h , u >. ) = <. ( g o. h ) , ( t A u ) >. ) |
34 |
30 31 32 33
|
syl3anc |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( <. g , t >. ( +g ` U ) <. h , u >. ) = <. ( g o. h ) , ( t A u ) >. ) |
35 |
34
|
eqeq2d |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) <-> <. F , S >. = <. ( g o. h ) , ( t A u ) >. ) ) |
36 |
|
vex |
|- g e. _V |
37 |
|
vex |
|- h e. _V |
38 |
36 37
|
coex |
|- ( g o. h ) e. _V |
39 |
|
ovex |
|- ( t A u ) e. _V |
40 |
38 39
|
opth2 |
|- ( <. F , S >. = <. ( g o. h ) , ( t A u ) >. <-> ( F = ( g o. h ) /\ S = ( t A u ) ) ) |
41 |
35 40
|
bitrdi |
|- ( ( ph /\ ( ( g e. T /\ t e. E ) /\ ( h e. T /\ u e. E ) ) ) -> ( <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) <-> ( F = ( g o. h ) /\ S = ( t A u ) ) ) ) |
42 |
29 41
|
syldan |
|- ( ( ph /\ ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) ) -> ( <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) <-> ( F = ( g o. h ) /\ S = ( t A u ) ) ) ) |
43 |
42
|
pm5.32da |
|- ( ph -> ( ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) ) <-> ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ ( F = ( g o. h ) /\ S = ( t A u ) ) ) ) ) |
44 |
43
|
4exbidv |
|- ( ph -> ( E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ <. F , S >. = ( <. g , t >. ( +g ` U ) <. h , u >. ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ ( F = ( g o. h ) /\ S = ( t A u ) ) ) ) ) |
45 |
20 44
|
bitrd |
|- ( ph -> ( <. F , S >. e. ( ( I ` X ) .(+) ( I ` Y ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` X ) /\ <. h , u >. e. ( I ` Y ) ) /\ ( F = ( g o. h ) /\ S = ( t A u ) ) ) ) ) |