Step |
Hyp |
Ref |
Expression |
1 |
|
dihprrn.h |
|- H = ( LHyp ` K ) |
2 |
|
dihprrn.u |
|- U = ( ( DVecH ` K ) ` W ) |
3 |
|
dihprrn.v |
|- V = ( Base ` U ) |
4 |
|
dihprrn.n |
|- N = ( LSpan ` U ) |
5 |
|
dihprrn.i |
|- I = ( ( DIsoH ` K ) ` W ) |
6 |
|
dihprrn.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
7 |
|
dihprrn.x |
|- ( ph -> X e. V ) |
8 |
|
dihprrn.y |
|- ( ph -> Y e. V ) |
9 |
|
dihprrnlem1.l |
|- .<_ = ( le ` K ) |
10 |
|
dihprrnlem1.o |
|- .0. = ( 0g ` U ) |
11 |
|
dihprrnlem1.nz |
|- ( ph -> Y =/= .0. ) |
12 |
|
dihprrnlem1.x |
|- ( ph -> ( `' I ` ( N ` { X } ) ) .<_ W ) |
13 |
|
dihprrnlem1.y |
|- ( ph -> -. ( `' I ` ( N ` { Y } ) ) .<_ W ) |
14 |
|
df-pr |
|- { X , Y } = ( { X } u. { Y } ) |
15 |
14
|
fveq2i |
|- ( N ` { X , Y } ) = ( N ` ( { X } u. { Y } ) ) |
16 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
17 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
18 |
|
eqid |
|- ( Atoms ` K ) = ( Atoms ` K ) |
19 |
|
eqid |
|- ( LSSum ` U ) = ( LSSum ` U ) |
20 |
1 2 3 4 5
|
dihlsprn |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran I ) |
21 |
6 7 20
|
syl2anc |
|- ( ph -> ( N ` { X } ) e. ran I ) |
22 |
16 1 5
|
dihcnvcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) e. ran I ) -> ( `' I ` ( N ` { X } ) ) e. ( Base ` K ) ) |
23 |
6 21 22
|
syl2anc |
|- ( ph -> ( `' I ` ( N ` { X } ) ) e. ( Base ` K ) ) |
24 |
23 12
|
jca |
|- ( ph -> ( ( `' I ` ( N ` { X } ) ) e. ( Base ` K ) /\ ( `' I ` ( N ` { X } ) ) .<_ W ) ) |
25 |
18 1 2 3 10 4 5
|
dihlspsnat |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. V /\ Y =/= .0. ) -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) ) |
26 |
6 8 11 25
|
syl3anc |
|- ( ph -> ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) ) |
27 |
26 13
|
jca |
|- ( ph -> ( ( `' I ` ( N ` { Y } ) ) e. ( Atoms ` K ) /\ -. ( `' I ` ( N ` { Y } ) ) .<_ W ) ) |
28 |
16 9 1 17 18 2 19 5 6 24 27
|
dihjatc |
|- ( ph -> ( I ` ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) = ( ( I ` ( `' I ` ( N ` { X } ) ) ) ( LSSum ` U ) ( I ` ( `' I ` ( N ` { Y } ) ) ) ) ) |
29 |
1 5
|
dihcnvid2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) e. ran I ) -> ( I ` ( `' I ` ( N ` { X } ) ) ) = ( N ` { X } ) ) |
30 |
6 21 29
|
syl2anc |
|- ( ph -> ( I ` ( `' I ` ( N ` { X } ) ) ) = ( N ` { X } ) ) |
31 |
1 2 3 4 5
|
dihlsprn |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. V ) -> ( N ` { Y } ) e. ran I ) |
32 |
6 8 31
|
syl2anc |
|- ( ph -> ( N ` { Y } ) e. ran I ) |
33 |
1 5
|
dihcnvid2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { Y } ) e. ran I ) -> ( I ` ( `' I ` ( N ` { Y } ) ) ) = ( N ` { Y } ) ) |
34 |
6 32 33
|
syl2anc |
|- ( ph -> ( I ` ( `' I ` ( N ` { Y } ) ) ) = ( N ` { Y } ) ) |
35 |
30 34
|
oveq12d |
|- ( ph -> ( ( I ` ( `' I ` ( N ` { X } ) ) ) ( LSSum ` U ) ( I ` ( `' I ` ( N ` { Y } ) ) ) ) = ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) ) |
36 |
1 2 6
|
dvhlmod |
|- ( ph -> U e. LMod ) |
37 |
7
|
snssd |
|- ( ph -> { X } C_ V ) |
38 |
8
|
snssd |
|- ( ph -> { Y } C_ V ) |
39 |
3 4 19
|
lsmsp2 |
|- ( ( U e. LMod /\ { X } C_ V /\ { Y } C_ V ) -> ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` ( { X } u. { Y } ) ) ) |
40 |
36 37 38 39
|
syl3anc |
|- ( ph -> ( ( N ` { X } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` ( { X } u. { Y } ) ) ) |
41 |
28 35 40
|
3eqtrrd |
|- ( ph -> ( N ` ( { X } u. { Y } ) ) = ( I ` ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) |
42 |
15 41
|
syl5eq |
|- ( ph -> ( N ` { X , Y } ) = ( I ` ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) ) |
43 |
6
|
simpld |
|- ( ph -> K e. HL ) |
44 |
43
|
hllatd |
|- ( ph -> K e. Lat ) |
45 |
16 1 5
|
dihcnvcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( N ` { Y } ) e. ran I ) -> ( `' I ` ( N ` { Y } ) ) e. ( Base ` K ) ) |
46 |
6 32 45
|
syl2anc |
|- ( ph -> ( `' I ` ( N ` { Y } ) ) e. ( Base ` K ) ) |
47 |
16 17
|
latjcl |
|- ( ( K e. Lat /\ ( `' I ` ( N ` { X } ) ) e. ( Base ` K ) /\ ( `' I ` ( N ` { Y } ) ) e. ( Base ` K ) ) -> ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) e. ( Base ` K ) ) |
48 |
44 23 46 47
|
syl3anc |
|- ( ph -> ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) e. ( Base ` K ) ) |
49 |
16 1 5
|
dihcl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) e. ( Base ` K ) ) -> ( I ` ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) e. ran I ) |
50 |
6 48 49
|
syl2anc |
|- ( ph -> ( I ` ( ( `' I ` ( N ` { X } ) ) ( join ` K ) ( `' I ` ( N ` { Y } ) ) ) ) e. ran I ) |
51 |
42 50
|
eqeltrd |
|- ( ph -> ( N ` { X , Y } ) e. ran I ) |