Step |
Hyp |
Ref |
Expression |
1 |
|
ltrnu.l |
|- .<_ = ( le ` K ) |
2 |
|
ltrnu.j |
|- .\/ = ( join ` K ) |
3 |
|
ltrnu.m |
|- ./\ = ( meet ` K ) |
4 |
|
ltrnu.a |
|- A = ( Atoms ` K ) |
5 |
|
ltrnu.h |
|- H = ( LHyp ` K ) |
6 |
|
ltrnu.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
an4 |
|- ( ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) <-> ( ( P e. A /\ Q e. A ) /\ ( -. P .<_ W /\ -. Q .<_ W ) ) ) |
8 |
|
simpr |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> ( P e. A /\ Q e. A ) ) |
9 |
|
simplr |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> F e. T ) |
10 |
|
eqid |
|- ( ( LDil ` K ) ` W ) = ( ( LDil ` K ) ` W ) |
11 |
1 2 3 4 5 10 6
|
isltrn |
|- ( ( K e. V /\ W e. H ) -> ( F e. T <-> ( F e. ( ( LDil ` K ) ` W ) /\ A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) ) |
12 |
11
|
ad2antrr |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> ( F e. T <-> ( F e. ( ( LDil ` K ) ` W ) /\ A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) ) |
13 |
|
simpr |
|- ( ( F e. ( ( LDil ` K ) ` W ) /\ A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) -> A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) |
14 |
12 13
|
syl6bi |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> ( F e. T -> A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) |
15 |
9 14
|
mpd |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) |
16 |
|
breq1 |
|- ( p = P -> ( p .<_ W <-> P .<_ W ) ) |
17 |
16
|
notbid |
|- ( p = P -> ( -. p .<_ W <-> -. P .<_ W ) ) |
18 |
17
|
anbi1d |
|- ( p = P -> ( ( -. p .<_ W /\ -. q .<_ W ) <-> ( -. P .<_ W /\ -. q .<_ W ) ) ) |
19 |
|
id |
|- ( p = P -> p = P ) |
20 |
|
fveq2 |
|- ( p = P -> ( F ` p ) = ( F ` P ) ) |
21 |
19 20
|
oveq12d |
|- ( p = P -> ( p .\/ ( F ` p ) ) = ( P .\/ ( F ` P ) ) ) |
22 |
21
|
oveq1d |
|- ( p = P -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( P .\/ ( F ` P ) ) ./\ W ) ) |
23 |
22
|
eqeq1d |
|- ( p = P -> ( ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) <-> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) |
24 |
18 23
|
imbi12d |
|- ( p = P -> ( ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) <-> ( ( -. P .<_ W /\ -. q .<_ W ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) ) ) |
25 |
|
breq1 |
|- ( q = Q -> ( q .<_ W <-> Q .<_ W ) ) |
26 |
25
|
notbid |
|- ( q = Q -> ( -. q .<_ W <-> -. Q .<_ W ) ) |
27 |
26
|
anbi2d |
|- ( q = Q -> ( ( -. P .<_ W /\ -. q .<_ W ) <-> ( -. P .<_ W /\ -. Q .<_ W ) ) ) |
28 |
|
id |
|- ( q = Q -> q = Q ) |
29 |
|
fveq2 |
|- ( q = Q -> ( F ` q ) = ( F ` Q ) ) |
30 |
28 29
|
oveq12d |
|- ( q = Q -> ( q .\/ ( F ` q ) ) = ( Q .\/ ( F ` Q ) ) ) |
31 |
30
|
oveq1d |
|- ( q = Q -> ( ( q .\/ ( F ` q ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |
32 |
31
|
eqeq2d |
|- ( q = Q -> ( ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) <-> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) ) |
33 |
27 32
|
imbi12d |
|- ( q = Q -> ( ( ( -. P .<_ W /\ -. q .<_ W ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) <-> ( ( -. P .<_ W /\ -. Q .<_ W ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) ) ) |
34 |
24 33
|
rspc2v |
|- ( ( P e. A /\ Q e. A ) -> ( A. p e. A A. q e. A ( ( -. p .<_ W /\ -. q .<_ W ) -> ( ( p .\/ ( F ` p ) ) ./\ W ) = ( ( q .\/ ( F ` q ) ) ./\ W ) ) -> ( ( -. P .<_ W /\ -. Q .<_ W ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) ) ) |
35 |
8 15 34
|
sylc |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ Q e. A ) ) -> ( ( -. P .<_ W /\ -. Q .<_ W ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) ) |
36 |
35
|
impr |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( ( P e. A /\ Q e. A ) /\ ( -. P .<_ W /\ -. Q .<_ W ) ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |
37 |
7 36
|
sylan2b |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |
38 |
37
|
3impb |
|- ( ( ( ( K e. V /\ W e. H ) /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( P .\/ ( F ` P ) ) ./\ W ) = ( ( Q .\/ ( F ` Q ) ) ./\ W ) ) |