Step |
Hyp |
Ref |
Expression |
1 |
|
lcfrlem17.h |
|- H = ( LHyp ` K ) |
2 |
|
lcfrlem17.o |
|- ._|_ = ( ( ocH ` K ) ` W ) |
3 |
|
lcfrlem17.u |
|- U = ( ( DVecH ` K ) ` W ) |
4 |
|
lcfrlem17.v |
|- V = ( Base ` U ) |
5 |
|
lcfrlem17.p |
|- .+ = ( +g ` U ) |
6 |
|
lcfrlem17.z |
|- .0. = ( 0g ` U ) |
7 |
|
lcfrlem17.n |
|- N = ( LSpan ` U ) |
8 |
|
lcfrlem17.a |
|- A = ( LSAtoms ` U ) |
9 |
|
lcfrlem17.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
10 |
|
lcfrlem17.x |
|- ( ph -> X e. ( V \ { .0. } ) ) |
11 |
|
lcfrlem17.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
12 |
|
lcfrlem17.ne |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
13 |
|
df-pr |
|- { X , Y } = ( { X } u. { Y } ) |
14 |
13
|
fveq2i |
|- ( ._|_ ` { X , Y } ) = ( ._|_ ` ( { X } u. { Y } ) ) |
15 |
10
|
eldifad |
|- ( ph -> X e. V ) |
16 |
15
|
snssd |
|- ( ph -> { X } C_ V ) |
17 |
11
|
eldifad |
|- ( ph -> Y e. V ) |
18 |
17
|
snssd |
|- ( ph -> { Y } C_ V ) |
19 |
1 3 4 2
|
dochdmj1 |
|- ( ( ( K e. HL /\ W e. H ) /\ { X } C_ V /\ { Y } C_ V ) -> ( ._|_ ` ( { X } u. { Y } ) ) = ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) ) |
20 |
9 16 18 19
|
syl3anc |
|- ( ph -> ( ._|_ ` ( { X } u. { Y } ) ) = ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) ) |
21 |
14 20
|
syl5eq |
|- ( ph -> ( ._|_ ` { X , Y } ) = ( ( ._|_ ` { X } ) i^i ( ._|_ ` { Y } ) ) ) |