Metamath Proof Explorer


Theorem dia2dimlem11

Description: Lemma for dia2dim . Convert ordering hypothesis on RF to subspace membership F e. ( I( U .\/ V ) ) . (Contributed by NM, 8-Sep-2014)

Ref Expression
Hypotheses dia2dimlem11.l
|- .<_ = ( le ` K )
dia2dimlem11.j
|- .\/ = ( join ` K )
dia2dimlem11.m
|- ./\ = ( meet ` K )
dia2dimlem11.a
|- A = ( Atoms ` K )
dia2dimlem11.h
|- H = ( LHyp ` K )
dia2dimlem11.t
|- T = ( ( LTrn ` K ) ` W )
dia2dimlem11.r
|- R = ( ( trL ` K ) ` W )
dia2dimlem11.y
|- Y = ( ( DVecA ` K ) ` W )
dia2dimlem11.s
|- S = ( LSubSp ` Y )
dia2dimlem11.pl
|- .(+) = ( LSSum ` Y )
dia2dimlem11.n
|- N = ( LSpan ` Y )
dia2dimlem11.i
|- I = ( ( DIsoA ` K ) ` W )
dia2dimlem11.k
|- ( ph -> ( K e. HL /\ W e. H ) )
dia2dimlem11.u
|- ( ph -> ( U e. A /\ U .<_ W ) )
dia2dimlem11.v
|- ( ph -> ( V e. A /\ V .<_ W ) )
dia2dimlem11.f
|- ( ph -> F e. T )
dia2dimlem11.uv
|- ( ph -> U =/= V )
dia2dimlem11.fe
|- ( ph -> F e. ( I ` ( U .\/ V ) ) )
Assertion dia2dimlem11
|- ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )

Proof

Step Hyp Ref Expression
1 dia2dimlem11.l
 |-  .<_ = ( le ` K )
2 dia2dimlem11.j
 |-  .\/ = ( join ` K )
3 dia2dimlem11.m
 |-  ./\ = ( meet ` K )
4 dia2dimlem11.a
 |-  A = ( Atoms ` K )
5 dia2dimlem11.h
 |-  H = ( LHyp ` K )
6 dia2dimlem11.t
 |-  T = ( ( LTrn ` K ) ` W )
7 dia2dimlem11.r
 |-  R = ( ( trL ` K ) ` W )
8 dia2dimlem11.y
 |-  Y = ( ( DVecA ` K ) ` W )
9 dia2dimlem11.s
 |-  S = ( LSubSp ` Y )
10 dia2dimlem11.pl
 |-  .(+) = ( LSSum ` Y )
11 dia2dimlem11.n
 |-  N = ( LSpan ` Y )
12 dia2dimlem11.i
 |-  I = ( ( DIsoA ` K ) ` W )
13 dia2dimlem11.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
14 dia2dimlem11.u
 |-  ( ph -> ( U e. A /\ U .<_ W ) )
15 dia2dimlem11.v
 |-  ( ph -> ( V e. A /\ V .<_ W ) )
16 dia2dimlem11.f
 |-  ( ph -> F e. T )
17 dia2dimlem11.uv
 |-  ( ph -> U =/= V )
18 dia2dimlem11.fe
 |-  ( ph -> F e. ( I ` ( U .\/ V ) ) )
19 1 2 4 5 6 7 8 9 11 12 13 14 15 16 18 dia2dimlem10
 |-  ( ph -> ( R ` F ) .<_ ( U .\/ V ) )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 19 17 dia2dimlem9
 |-  ( ph -> F e. ( ( I ` U ) .(+) ( I ` V ) ) )