Step |
Hyp |
Ref |
Expression |
1 |
|
dia2dimlem6.l |
|- .<_ = ( le ` K ) |
2 |
|
dia2dimlem6.j |
|- .\/ = ( join ` K ) |
3 |
|
dia2dimlem6.m |
|- ./\ = ( meet ` K ) |
4 |
|
dia2dimlem6.a |
|- A = ( Atoms ` K ) |
5 |
|
dia2dimlem6.h |
|- H = ( LHyp ` K ) |
6 |
|
dia2dimlem6.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
dia2dimlem6.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
dia2dimlem6.y |
|- Y = ( ( DVecA ` K ) ` W ) |
9 |
|
dia2dimlem6.s |
|- S = ( LSubSp ` Y ) |
10 |
|
dia2dimlem6.pl |
|- .(+) = ( LSSum ` Y ) |
11 |
|
dia2dimlem6.n |
|- N = ( LSpan ` Y ) |
12 |
|
dia2dimlem6.i |
|- I = ( ( DIsoA ` K ) ` W ) |
13 |
|
dia2dimlem6.q |
|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) ) |
14 |
|
dia2dimlem6.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
15 |
|
dia2dimlem6.u |
|- ( ph -> ( U e. A /\ U .<_ W ) ) |
16 |
|
dia2dimlem6.v |
|- ( ph -> ( V e. A /\ V .<_ W ) ) |
17 |
|
dia2dimlem6.p |
|- ( ph -> ( P e. A /\ -. P .<_ W ) ) |
18 |
|
dia2dimlem6.f |
|- ( ph -> ( F e. T /\ ( F ` P ) =/= P ) ) |
19 |
|
dia2dimlem6.rf |
|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) ) |
20 |
|
dia2dimlem6.uv |
|- ( ph -> U =/= V ) |
21 |
|
dia2dimlem6.ru |
|- ( ph -> ( R ` F ) =/= U ) |
22 |
|
dia2dimlem6.rv |
|- ( ph -> ( R ` F ) =/= V ) |
23 |
1 2 3 4 5 6 7 13 14 15 16 17 18 19 20 21
|
dia2dimlem1 |
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) ) |
24 |
18
|
simpld |
|- ( ph -> F e. T ) |
25 |
1 4 5 6
|
ltrnel |
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) |
26 |
14 24 17 25
|
syl3anc |
|- ( ph -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) |
27 |
1 4 5 6
|
cdleme50ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> E. d e. T ( d ` Q ) = ( F ` P ) ) |
28 |
14 23 26 27
|
syl3anc |
|- ( ph -> E. d e. T ( d ` Q ) = ( F ` P ) ) |
29 |
1 4 5 6
|
cdleme50ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> E. g e. T ( g ` P ) = Q ) |
30 |
14 17 23 29
|
syl3anc |
|- ( ph -> E. g e. T ( g ` P ) = Q ) |
31 |
14
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( K e. HL /\ W e. H ) ) |
32 |
15
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( U e. A /\ U .<_ W ) ) |
33 |
16
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( V e. A /\ V .<_ W ) ) |
34 |
17
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( P e. A /\ -. P .<_ W ) ) |
35 |
18
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( F e. T /\ ( F ` P ) =/= P ) ) |
36 |
19
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) .<_ ( U .\/ V ) ) |
37 |
20
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> U =/= V ) |
38 |
21
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) =/= U ) |
39 |
22
|
3ad2ant1 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( R ` F ) =/= V ) |
40 |
|
simp21 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> g e. T ) |
41 |
|
simp22 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( g ` P ) = Q ) |
42 |
|
simp23 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> d e. T ) |
43 |
|
simp3 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> ( d ` Q ) = ( F ` P ) ) |
44 |
1 2 3 4 5 6 7 8 9 10 11 12 13 31 32 33 34 35 36 37 38 39 40 41 42 43
|
dia2dimlem5 |
|- ( ( ph /\ ( g e. T /\ ( g ` P ) = Q /\ d e. T ) /\ ( d ` Q ) = ( F ` P ) ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) |
45 |
44
|
3exp |
|- ( ph -> ( ( g e. T /\ ( g ` P ) = Q /\ d e. T ) -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) |
46 |
45
|
3expd |
|- ( ph -> ( g e. T -> ( ( g ` P ) = Q -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) ) ) |
47 |
46
|
rexlimdv |
|- ( ph -> ( E. g e. T ( g ` P ) = Q -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) ) |
48 |
30 47
|
mpd |
|- ( ph -> ( d e. T -> ( ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) ) |
49 |
48
|
rexlimdv |
|- ( ph -> ( E. d e. T ( d ` Q ) = ( F ` P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) ) |
50 |
28 49
|
mpd |
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) |