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Metamath Proof Explorer

Theorem cdlemn5pre

Description: Part of proof of Lemma N of Crawley p. 121 line 32. (Contributed by NM, 25-Feb-2014)

Ref Expression
Hypotheses cdlemn5.b 
|- B = ( Base ` K )
cdlemn5.l 
|- .<_ = ( le ` K )
cdlemn5.j 
|- .\/ = ( join ` K )
cdlemn5.a 
|- A = ( Atoms ` K )
cdlemn5.h 
|- H = ( LHyp ` K )
cdlemn5.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn5.s 
|- .(+) = ( LSSum ` U )
cdlemn5.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn5.J 
|- J = ( ( DIsoC ` K ) ` W )
cdlemn5.p 
|- P = ( ( oc ` K ) ` W )
cdlemn5.o 
|- O = ( h e. T |-> ( _I |` B ) )
cdlemn5.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemn5.e 
|- E = ( ( TEndo ` K ) ` W )
cdlemn5.n 
|- N = ( LSpan ` U )
cdlemn5.f 
|- F = ( iota_ h e. T ( h ` P ) = Q )
cdlemn5.g 
|- G = ( iota_ h e. T ( h ` P ) = R )
cdlemn5.m 
|- M = ( iota_ h e. T ( h ` Q ) = R )
Assertion cdlemn5pre 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )

Proof

Step Hyp Ref Expression
1 cdlemn5.b 
 |-  B = ( Base ` K )
2 cdlemn5.l 
 |-  .<_ = ( le ` K )
3 cdlemn5.j 
 |-  .\/ = ( join ` K )
4 cdlemn5.a 
 |-  A = ( Atoms ` K )
5 cdlemn5.h 
 |-  H = ( LHyp ` K )
6 cdlemn5.u 
 |-  U = ( ( DVecH ` K ) ` W )
7 cdlemn5.s 
 |-  .(+) = ( LSSum ` U )
8 cdlemn5.i 
 |-  I = ( ( DIsoB ` K ) ` W )
9 cdlemn5.J 
 |-  J = ( ( DIsoC ` K ) ` W )
10 cdlemn5.p 
 |-  P = ( ( oc ` K ) ` W )
11 cdlemn5.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
12 cdlemn5.t 
 |-  T = ( ( LTrn ` K ) ` W )
13 cdlemn5.e 
 |-  E = ( ( TEndo ` K ) ` W )
14 cdlemn5.n 
 |-  N = ( LSpan ` U )
15 cdlemn5.f 
 |-  F = ( iota_ h e. T ( h ` P ) = Q )
16 cdlemn5.g 
 |-  G = ( iota_ h e. T ( h ` P ) = R )
17 cdlemn5.m 
 |-  M = ( iota_ h e. T ( h ` Q ) = R )
18 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( K e. HL /\ W e. H ) )
19 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( R e. A /\ -. R .<_ W ) )
20 2 4 5 10 12 9 6 14 16 diclspsn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( J ` R ) = ( N ` { <. G , ( _I |` T ) >. } ) )
21 18 19 20 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` R ) = ( N ` { <. G , ( _I |` T ) >. } ) )
22 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( Q e. A /\ -. Q .<_ W ) )
23 1 2 4 10 5 12 11 6 15 16 17 14 7 cdlemn4a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( N ` { <. G , ( _I |` T ) >. } ) C_ ( ( N ` { <. F , ( _I |` T ) >. } ) .(+) ( N ` { <. M , O >. } ) ) )
24 18 22 19 23 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( N ` { <. G , ( _I |` T ) >. } ) C_ ( ( N ` { <. F , ( _I |` T ) >. } ) .(+) ( N ` { <. M , O >. } ) ) )
25 2 4 5 10 12 9 6 14 15 diclspsn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( J ` Q ) = ( N ` { <. F , ( _I |` T ) >. } ) )
26 18 22 25 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` Q ) = ( N ` { <. F , ( _I |` T ) >. } ) )
27 26 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( ( J ` Q ) .(+) ( N ` { <. M , O >. } ) ) = ( ( N ` { <. F , ( _I |` T ) >. } ) .(+) ( N ` { <. M , O >. } ) ) )
28 24 27 sseqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( N ` { <. G , ( _I |` T ) >. } ) C_ ( ( J ` Q ) .(+) ( N ` { <. M , O >. } ) ) )
29 5 6 18 dvhlmod 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> U e. LMod )
30 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
31 30 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
32 29 31 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
33 2 4 5 6 9 30 diclss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
34 18 22 33 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
35 32 34 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` Q ) e. ( SubGrp ` U ) )
36 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( X e. B /\ X .<_ W ) )
37 1 2 5 6 8 30 diblss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) e. ( LSubSp ` U ) )
38 18 36 37 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( I ` X ) e. ( LSubSp ` U ) )
39 32 38 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( I ` X ) e. ( SubGrp ` U ) )
40 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
41 1 2 3 4 5 12 40 11 8 6 14 17 cdlemn2a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( N ` { <. M , O >. } ) C_ ( I ` X ) )
42 7 lsmless2 
 |-  ( ( ( J ` Q ) e. ( SubGrp ` U ) /\ ( I ` X ) e. ( SubGrp ` U ) /\ ( N ` { <. M , O >. } ) C_ ( I ` X ) ) -> ( ( J ` Q ) .(+) ( N ` { <. M , O >. } ) ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )
43 35 39 41 42 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( ( J ` Q ) .(+) ( N ` { <. M , O >. } ) ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )
44 28 43 sstrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( N ` { <. G , ( _I |` T ) >. } ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )
45 21 44 eqsstrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )