Metamath Proof Explorer

Theorem cdlemn2a

Description: Part of proof of Lemma N of Crawley p. 121. (Contributed by NM, 24-Feb-2014)

Ref Expression
Hypotheses cdlemn2a.b 
|- B = ( Base ` K )
cdlemn2a.l 
|- .<_ = ( le ` K )
cdlemn2a.j 
|- .\/ = ( join ` K )
cdlemn2a.a 
|- A = ( Atoms ` K )
cdlemn2a.h 
|- H = ( LHyp ` K )
cdlemn2a.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemn2a.r 
|- R = ( ( trL ` K ) ` W )
cdlemn2a.o 
|- O = ( f e. T |-> ( _I |` B ) )
cdlemn2a.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn2a.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn2a.n 
|- N = ( LSpan ` U )
cdlemn2a.f 
|- F = ( iota_ h e. T ( h ` Q ) = S )
Assertion cdlemn2a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( N ` { <. F , O >. } ) C_ ( I ` X ) )

Proof

Step Hyp Ref Expression
1 cdlemn2a.b 
 |-  B = ( Base ` K )
2 cdlemn2a.l 
 |-  .<_ = ( le ` K )
3 cdlemn2a.j 
 |-  .\/ = ( join ` K )
4 cdlemn2a.a 
 |-  A = ( Atoms ` K )
5 cdlemn2a.h 
 |-  H = ( LHyp ` K )
6 cdlemn2a.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemn2a.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemn2a.o 
 |-  O = ( f e. T |-> ( _I |` B ) )
9 cdlemn2a.i 
 |-  I = ( ( DIsoB ` K ) ` W )
10 cdlemn2a.u 
 |-  U = ( ( DVecH ` K ) ` W )
11 cdlemn2a.n 
 |-  N = ( LSpan ` U )
12 cdlemn2a.f 
 |-  F = ( iota_ h e. T ( h ` Q ) = S )
13 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( K e. HL /\ W e. H ) )
14 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( Q e. A /\ -. Q .<_ W ) )
15 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( S e. A /\ -. S .<_ W ) )
16 2 4 5 6 12 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> F e. T )
17 13 14 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> F e. T )
18 1 5 6 7 8 10 9 11 dib1dim2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) ) = ( N ` { <. F , O >. } ) )
19 13 17 18 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( I ` ( R ` F ) ) = ( N ` { <. F , O >. } ) )
20 1 2 3 4 5 6 7 12 cdlemn2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( R ` F ) .<_ X )
21 1 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. B )
22 13 17 21 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( R ` F ) e. B )
23 2 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
24 13 17 23 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( R ` F ) .<_ W )
25 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( X e. B /\ X .<_ W ) )
26 1 2 5 9 dibord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( R ` F ) e. B /\ ( R ` F ) .<_ W ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( I ` ( R ` F ) ) C_ ( I ` X ) <-> ( R ` F ) .<_ X ) )
27 13 22 24 25 26 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( ( I ` ( R ` F ) ) C_ ( I ` X ) <-> ( R ` F ) .<_ X ) )
28 20 27 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( I ` ( R ` F ) ) C_ ( I ` X ) )
29 19 28 eqsstrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ S .<_ ( Q .\/ X ) ) -> ( N ` { <. F , O >. } ) C_ ( I ` X ) )