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Metamath Proof Explorer

Theorem cdlemn11a

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 cdlemn11a 
|- ( ( ( 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 ` N ) )

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 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 ) )
19 2 4 5 6 lhpocnel2 
 |-  ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) )
20 19 3ad2ant1 
 |-  ( ( ( 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 ) ) ) -> ( P e. A /\ -. P .<_ W ) )
21 simp22 
 |-  ( ( ( 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 ) ) ) -> ( N e. A /\ -. N .<_ W ) )
22 2 4 5 8 17 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) -> G e. T )
23 18 20 21 22 syl3anc 
 |-  ( ( ( 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 e. T )
24 fvresi 
 |-  ( G e. T -> ( ( _I |` T ) ` G ) = G )
25 23 24 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 ) ) ) -> ( ( _I |` T ) ` G ) = G )
26 25 eqcomd 
 |-  ( ( ( 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 ) ` G ) )
27 5 8 10 tendoidcl 
 |-  ( ( K e. HL /\ W e. H ) -> ( _I |` T ) e. E )
28 27 3ad2ant1 
 |-  ( ( ( 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 |` T ) e. E )
29 riotaex 
 |-  ( iota_ h e. T ( h ` P ) = N ) e. _V
30 17 29 eqeltri 
 |-  G e. _V
31 8 fvexi 
 |-  T e. _V
32 resiexg 
 |-  ( T e. _V -> ( _I |` T ) e. _V )
33 31 32 ax-mp 
 |-  ( _I |` T ) e. _V
34 2 4 5 6 8 10 12 17 30 33 dicopelval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N e. A /\ -. N .<_ W ) ) -> ( <. G , ( _I |` T ) >. e. ( J ` N ) <-> ( G = ( ( _I |` T ) ` G ) /\ ( _I |` T ) e. E ) ) )
35 18 21 34 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 ` N ) <-> ( G = ( ( _I |` T ) ` G ) /\ ( _I |` T ) e. E ) ) )
36 26 28 35 mpbir2and 
 |-  ( ( ( 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 ` N ) )