Metamath Proof Explorer

Theorem cdlemn11c

Description: Part of proof of Lemma N of Crawley p. 121 line 37. (Contributed by NM, 27-Feb-2014)

Ref Expression
Hypotheses cdlemn11a.b 
|- B = ( Base ` K )
cdlemn11a.l 
|- .<_ = ( le ` K )
cdlemn11a.j 
|- .\/ = ( join ` K )
cdlemn11a.a 
|- A = ( Atoms ` K )
cdlemn11a.h 
|- H = ( LHyp ` K )
cdlemn11a.p 
|- P = ( ( oc ` K ) ` W )
cdlemn11a.o 
|- O = ( h e. T |-> ( _I |` B ) )
cdlemn11a.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemn11a.r 
|- R = ( ( trL ` K ) ` W )
cdlemn11a.e 
|- E = ( ( TEndo ` K ) ` W )
cdlemn11a.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn11a.J 
|- J = ( ( DIsoC ` K ) ` W )
cdlemn11a.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn11a.d 
|- .+ = ( +g ` U )
cdlemn11a.s 
|- .(+) = ( LSSum ` U )
cdlemn11a.f 
|- F = ( iota_ h e. T ( h ` P ) = Q )
cdlemn11a.g 
|- G = ( iota_ h e. T ( h ` P ) = N )
Assertion cdlemn11c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) )

Proof

Step Hyp Ref Expression
1 cdlemn11a.b 
 |-  B = ( Base ` K )
2 cdlemn11a.l 
 |-  .<_ = ( le ` K )
3 cdlemn11a.j 
 |-  .\/ = ( join ` K )
4 cdlemn11a.a 
 |-  A = ( Atoms ` K )
5 cdlemn11a.h 
 |-  H = ( LHyp ` K )
6 cdlemn11a.p 
 |-  P = ( ( oc ` K ) ` W )
7 cdlemn11a.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
8 cdlemn11a.t 
 |-  T = ( ( LTrn ` K ) ` W )
9 cdlemn11a.r 
 |-  R = ( ( trL ` K ) ` W )
10 cdlemn11a.e 
 |-  E = ( ( TEndo ` K ) ` W )
11 cdlemn11a.i 
 |-  I = ( ( DIsoB ` K ) ` W )
12 cdlemn11a.J 
 |-  J = ( ( DIsoC ` K ) ` W )
13 cdlemn11a.u 
 |-  U = ( ( DVecH ` K ) ` W )
14 cdlemn11a.d 
 |-  .+ = ( +g ` U )
15 cdlemn11a.s 
 |-  .(+) = ( LSSum ` U )
16 cdlemn11a.f 
 |-  F = ( iota_ h e. T ( h ` P ) = Q )
17 cdlemn11a.g 
 |-  G = ( iota_ h e. T ( h ` P ) = N )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 cdlemn11b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> <. G , ( _I |` T ) >. e. ( ( J ` Q ) .(+) ( I ` X ) ) )
19 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( K e. HL /\ W e. H ) )
20 5 13 19 dvhlmod 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> U e. LMod )
21 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
22 21 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
23 20 22 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
24 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
25 2 4 5 13 12 21 diclss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
26 19 24 25 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( J ` Q ) e. ( LSubSp ` U ) )
27 23 26 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( J ` Q ) e. ( SubGrp ` U ) )
28 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> X e. B )
29 simp23r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> X .<_ W )
30 1 2 5 13 11 21 diblss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) e. ( LSubSp ` U ) )
31 19 28 29 30 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( I ` X ) e. ( LSubSp ` U ) )
32 23 31 sseldd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( I ` X ) e. ( SubGrp ` U ) )
33 14 15 lsmelval 
 |-  ( ( ( J ` Q ) e. ( SubGrp ` U ) /\ ( I ` X ) e. ( SubGrp ` U ) ) -> ( <. G , ( _I |` T ) >. e. ( ( J ` Q ) .(+) ( I ` X ) ) <-> E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) ) )
34 27 32 33 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> ( <. G , ( _I |` T ) >. e. ( ( J ` Q ) .(+) ( I ` X ) ) <-> E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) ) )
35 18 34 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` N ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> E. y e. ( J ` Q ) E. z e. ( I ` X ) <. G , ( _I |` T ) >. = ( y .+ z ) )