Metamath Proof Explorer

Theorem dia2dimlem7

Description: Lemma for dia2dim . Eliminate ( FP ) =/= P condition. (Contributed by NM, 8-Sep-2014)

Ref Expression
Hypotheses dia2dimlem7.l 
|- .<_ = ( le ` K )
dia2dimlem7.j 
|- .\/ = ( join ` K )
dia2dimlem7.m 
|- ./\ = ( meet ` K )
dia2dimlem7.a 
|- A = ( Atoms ` K )
dia2dimlem7.h 
|- H = ( LHyp ` K )
dia2dimlem7.t 
|- T = ( ( LTrn ` K ) ` W )
dia2dimlem7.r 
|- R = ( ( trL ` K ) ` W )
dia2dimlem7.y 
|- Y = ( ( DVecA ` K ) ` W )
dia2dimlem7.s 
|- S = ( LSubSp ` Y )
dia2dimlem7.pl 
|- .(+) = ( LSSum ` Y )
dia2dimlem7.n 
|- N = ( LSpan ` Y )
dia2dimlem7.i 
|- I = ( ( DIsoA ` K ) ` W )
dia2dimlem7.q 
|- Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) )
dia2dimlem7.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dia2dimlem7.u 
|- ( ph -> ( U e. A /\ U .<_ W ) )
dia2dimlem7.v 
|- ( ph -> ( V e. A /\ V .<_ W ) )
dia2dimlem7.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
dia2dimlem7.f 
|- ( ph -> F e. T )
dia2dimlem7.rf 
|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
dia2dimlem7.uv 
|- ( ph -> U =/= V )
dia2dimlem7.ru 
|- ( ph -> ( R ` F ) =/= U )
dia2dimlem7.rv 
|- ( ph -> ( R ` F ) =/= V )
Assertion dia2dimlem7 
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Proof

Step Hyp Ref Expression
1 dia2dimlem7.l 
 |-  .<_ = ( le ` K )
2 dia2dimlem7.j 
 |-  .\/ = ( join ` K )
3 dia2dimlem7.m 
 |-  ./\ = ( meet ` K )
4 dia2dimlem7.a 
 |-  A = ( Atoms ` K )
5 dia2dimlem7.h 
 |-  H = ( LHyp ` K )
6 dia2dimlem7.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 dia2dimlem7.r 
 |-  R = ( ( trL ` K ) ` W )
8 dia2dimlem7.y 
 |-  Y = ( ( DVecA ` K ) ` W )
9 dia2dimlem7.s 
 |-  S = ( LSubSp ` Y )
10 dia2dimlem7.pl 
 |-  .(+) = ( LSSum ` Y )
11 dia2dimlem7.n 
 |-  N = ( LSpan ` Y )
12 dia2dimlem7.i 
 |-  I = ( ( DIsoA ` K ) ` W )
13 dia2dimlem7.q 
 |-  Q = ( ( P .\/ U ) ./\ ( ( F ` P ) .\/ V ) )
14 dia2dimlem7.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 dia2dimlem7.u 
 |-  ( ph -> ( U e. A /\ U .<_ W ) )
16 dia2dimlem7.v 
 |-  ( ph -> ( V e. A /\ V .<_ W ) )
17 dia2dimlem7.p 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
18 dia2dimlem7.f 
 |-  ( ph -> F e. T )
19 dia2dimlem7.rf 
 |-  ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
20 dia2dimlem7.uv 
 |-  ( ph -> U =/= V )
21 dia2dimlem7.ru 
 |-  ( ph -> ( R ` F ) =/= U )
22 dia2dimlem7.rv 
 |-  ( ph -> ( R ` F ) =/= V )
23 eqid 
 |-  ( Base ` K ) = ( Base ` K )
24 23 1 4 5 6 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
25 14 18 17 24 syl3anc 
 |-  ( ph -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
26 eqid 
 |-  ( 0g ` Y ) = ( 0g ` Y )
27 23 5 6 8 26 dva0g 
 |-  ( ( K e. HL /\ W e. H ) -> ( 0g ` Y ) = ( _I |` ( Base ` K ) ) )
28 14 27 syl 
 |-  ( ph -> ( 0g ` Y ) = ( _I |` ( Base ` K ) ) )
29 5 8 dvalvec 
 |-  ( ( K e. HL /\ W e. H ) -> Y e. LVec )
30 lveclmod 
 |-  ( Y e. LVec -> Y e. LMod )
31 14 29 30 3syl 
 |-  ( ph -> Y e. LMod )
32 15 simpld 
 |-  ( ph -> U e. A )
33 23 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
34 32 33 syl 
 |-  ( ph -> U e. ( Base ` K ) )
35 15 simprd 
 |-  ( ph -> U .<_ W )
36 23 1 5 8 12 9 dialss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( U e. ( Base ` K ) /\ U .<_ W ) ) -> ( I ` U ) e. S )
37 14 34 35 36 syl12anc 
 |-  ( ph -> ( I ` U ) e. S )
38 16 simpld 
 |-  ( ph -> V e. A )
39 23 4 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
40 38 39 syl 
 |-  ( ph -> V e. ( Base ` K ) )
41 16 simprd 
 |-  ( ph -> V .<_ W )
42 23 1 5 8 12 9 dialss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( V e. ( Base ` K ) /\ V .<_ W ) ) -> ( I ` V ) e. S )
43 14 40 41 42 syl12anc 
 |-  ( ph -> ( I ` V ) e. S )
44 9 10 lsmcl 
 |-  ( ( Y e. LMod /\ ( I ` U ) e. S /\ ( I ` V ) e. S ) -> ( ( I ` U ) .(+) ( I ` V ) ) e. S )
45 31 37 43 44 syl3anc 
 |-  ( ph -> ( ( I ` U ) .(+) ( I ` V ) ) e. S )
46 26 9 lss0cl 
 |-  ( ( Y e. LMod /\ ( ( I ` U ) .(+) ( I ` V ) ) e. S ) -> ( 0g ` Y ) e. ( ( I ` U ) .(+) ( I ` V ) ) )
47 31 45 46 syl2anc 
 |-  ( ph -> ( 0g ` Y ) e. ( ( I ` U ) .(+) ( I ` V ) ) )
48 28 47 eqeltrrd 
 |-  ( ph -> ( _I |` ( Base ` K ) ) e. ( ( I ` U ) .(+) ( I ` V ) ) )
49 eleq1a 
 |-  ( ( _I |` ( Base ` K ) ) e. ( ( I ` U ) .(+) ( I ` V ) ) -> ( F = ( _I |` ( Base ` K ) ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) )
50 48 49 syl 
 |-  ( ph -> ( F = ( _I |` ( Base ` K ) ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) )
51 25 50 sylbird 
 |-  ( ph -> ( ( F ` P ) = P -> F e. ( ( I ` U ) .(+) ( I ` V ) ) ) )
52 51 imp 
 |-  ( ( ph /\ ( F ` P ) = P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )
53 14 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( K e. HL /\ W e. H ) )
54 15 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( U e. A /\ U .<_ W ) )
55 16 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( V e. A /\ V .<_ W ) )
56 17 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( P e. A /\ -. P .<_ W ) )
57 18 anim1i 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( F e. T /\ ( F ` P ) =/= P ) )
58 19 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( R ` F ) .<_ ( U .\/ V ) )
59 20 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> U =/= V )
60 21 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( R ` F ) =/= U )
61 22 adantr 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> ( R ` F ) =/= V )
62 1 2 3 4 5 6 7 8 9 10 11 12 13 53 54 55 56 57 58 59 60 61 dia2dimlem6 
 |-  ( ( ph /\ ( F ` P ) =/= P ) -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )
63 52 62 pm2.61dane 
 |-  ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )