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Metamath Proof Explorer

Theorem dia2dimlem8

Description: Lemma for dia2dim . Eliminate no-longer used auxiliary atoms P and Q . (Contributed by NM, 8-Sep-2014)

Ref Expression
Hypotheses dia2dimlem8.l 
|- .<_ = ( le ` K )
dia2dimlem8.j 
|- .\/ = ( join ` K )
dia2dimlem8.m 
|- ./\ = ( meet ` K )
dia2dimlem8.a 
|- A = ( Atoms ` K )
dia2dimlem8.h 
|- H = ( LHyp ` K )
dia2dimlem8.t 
|- T = ( ( LTrn ` K ) ` W )
dia2dimlem8.r 
|- R = ( ( trL ` K ) ` W )
dia2dimlem8.y 
|- Y = ( ( DVecA ` K ) ` W )
dia2dimlem8.s 
|- S = ( LSubSp ` Y )
dia2dimlem8.pl 
|- .(+) = ( LSSum ` Y )
dia2dimlem8.n 
|- N = ( LSpan ` Y )
dia2dimlem8.i 
|- I = ( ( DIsoA ` K ) ` W )
dia2dimlem8.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dia2dimlem8.u 
|- ( ph -> ( U e. A /\ U .<_ W ) )
dia2dimlem8.v 
|- ( ph -> ( V e. A /\ V .<_ W ) )
dia2dimlem8.f 
|- ( ph -> F e. T )
dia2dimlem8.rf 
|- ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
dia2dimlem8.uv 
|- ( ph -> U =/= V )
dia2dimlem8.ru 
|- ( ph -> ( R ` F ) =/= U )
dia2dimlem8.rv 
|- ( ph -> ( R ` F ) =/= V )
Assertion dia2dimlem8 
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Proof

Step Hyp Ref Expression
1 dia2dimlem8.l 
 |-  .<_ = ( le ` K )
2 dia2dimlem8.j 
 |-  .\/ = ( join ` K )
3 dia2dimlem8.m 
 |-  ./\ = ( meet ` K )
4 dia2dimlem8.a 
 |-  A = ( Atoms ` K )
5 dia2dimlem8.h 
 |-  H = ( LHyp ` K )
6 dia2dimlem8.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 dia2dimlem8.r 
 |-  R = ( ( trL ` K ) ` W )
8 dia2dimlem8.y 
 |-  Y = ( ( DVecA ` K ) ` W )
9 dia2dimlem8.s 
 |-  S = ( LSubSp ` Y )
10 dia2dimlem8.pl 
 |-  .(+) = ( LSSum ` Y )
11 dia2dimlem8.n 
 |-  N = ( LSpan ` Y )
12 dia2dimlem8.i 
 |-  I = ( ( DIsoA ` K ) ` W )
13 dia2dimlem8.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
14 dia2dimlem8.u 
 |-  ( ph -> ( U e. A /\ U .<_ W ) )
15 dia2dimlem8.v 
 |-  ( ph -> ( V e. A /\ V .<_ W ) )
16 dia2dimlem8.f 
 |-  ( ph -> F e. T )
17 dia2dimlem8.rf 
 |-  ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
18 dia2dimlem8.uv 
 |-  ( ph -> U =/= V )
19 dia2dimlem8.ru 
 |-  ( ph -> ( R ` F ) =/= U )
20 dia2dimlem8.rv 
 |-  ( ph -> ( R ` F ) =/= V )
21 eqid 
 |-  ( ( ( ( oc ` K ) ` W ) .\/ U ) ./\ ( ( F ` ( ( oc ` K ) ` W ) ) .\/ V ) ) = ( ( ( ( oc ` K ) ` W ) .\/ U ) ./\ ( ( F ` ( ( oc ` K ) ` W ) ) .\/ V ) )
22 eqid 
 |-  ( oc ` K ) = ( oc ` K )
23 1 22 4 5 lhpocnel 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
24 13 23 syl 
 |-  ( ph -> ( ( ( oc ` K ) ` W ) e. A /\ -. ( ( oc ` K ) ` W ) .<_ W ) )
25 1 2 3 4 5 6 7 8 9 10 11 12 21 13 14 15 24 16 17 18 19 20 dia2dimlem7 
 |-  ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )