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Metamath Proof Explorer

Theorem dia2dimlem6

Description: Lemma for dia2dim . Eliminate auxiliary translations G and D . (Contributed by NM, 8-Sep-2014)

Ref Expression
Hypotheses dia2dimlem6.l 
|- .<_ = ( le ` K )
dia2dimlem6.j 
|- .\/ = ( join ` K )
dia2dimlem6.m 
|- ./\ = ( meet ` K )
dia2dimlem6.a 
|- A = ( Atoms ` K )
dia2dimlem6.h 
|- H = ( LHyp ` K )
dia2dimlem6.t 
|- T = ( ( LTrn ` K ) ` W )
dia2dimlem6.r 
|- R = ( ( trL ` K ) ` W )
dia2dimlem6.y 
|- Y = ( ( DVecA ` K ) ` W )
dia2dimlem6.s 
|- S = ( LSubSp ` Y )
dia2dimlem6.pl 
|- .(+) = ( LSSum ` Y )
dia2dimlem6.n 
|- N = ( LSpan ` Y )
dia2dimlem6.i 
|- I = ( ( DIsoA ` K ) ` W )
dia2dimlem6.q 
|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) )
dia2dimlem6.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dia2dimlem6.u 
|- ( ph -> ( U e. A /\ U .<_ W ) )
dia2dimlem6.v 
|- ( ph -> ( V e. A /\ V .<_ W ) )
dia2dimlem6.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
dia2dimlem6.f 
|- ( ph -> ( F e. T /\ ( F ` P ) =/= P ) )
dia2dimlem6.rf 
|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
dia2dimlem6.uv 
|- ( ph -> U =/= V )
dia2dimlem6.ru 
|- ( ph -> ( R ` F ) =/= U )
dia2dimlem6.rv 
|- ( ph -> ( R ` F ) =/= V )
Assertion dia2dimlem6 
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Proof

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 ) ) )