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Metamath Proof Explorer

Theorem dihord2

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. TODO: do we need -. X .<_ W and -. Y .<_ W ? (Contributed by NM, 4-Mar-2014)

Ref Expression
Hypotheses dihord2.b 
|- B = ( Base ` K )
dihord2.l 
|- .<_ = ( le ` K )
dihord2.j 
|- .\/ = ( join ` K )
dihord2.m 
|- ./\ = ( meet ` K )
dihord2.a 
|- A = ( Atoms ` K )
dihord2.h 
|- H = ( LHyp ` K )
dihord2.i 
|- I = ( ( DIsoB ` K ) ` W )
dihord2.J 
|- J = ( ( DIsoC ` K ) ` W )
dihord2.u 
|- U = ( ( DVecH ` K ) ` W )
dihord2.s 
|- .(+) = ( LSSum ` U )
Assertion dihord2 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( X e. B /\ Y e. B ) /\ ( ( Q .\/ ( X ./\ W ) ) = X /\ ( N .\/ ( Y ./\ W ) ) = Y /\ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` N ) .(+) ( I ` ( Y ./\ W ) ) ) ) ) -> X .<_ Y )

Proof

Step Hyp Ref Expression
1 dihord2.b 
 |-  B = ( Base ` K )
2 dihord2.l 
 |-  .<_ = ( le ` K )
3 dihord2.j 
 |-  .\/ = ( join ` K )
4 dihord2.m 
 |-  ./\ = ( meet ` K )
5 dihord2.a 
 |-  A = ( Atoms ` K )
6 dihord2.h 
 |-  H = ( LHyp ` K )
7 dihord2.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 dihord2.J 
 |-  J = ( ( DIsoC ` K ) ` W )
9 dihord2.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 dihord2.s 
 |-  .(+) = ( LSSum ` U )
11 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
12 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
13 eqid 
 |-  ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
14 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
15 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16 eqid 
 |-  ( +g ` U ) = ( +g ` U )
17 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = N ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = N )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 dihord2pre2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( X e. B /\ Y e. B ) /\ ( ( Q .\/ ( X ./\ W ) ) = X /\ ( N .\/ ( Y ./\ W ) ) = Y /\ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` N ) .(+) ( I ` ( Y ./\ W ) ) ) ) ) -> ( Q .\/ ( X ./\ W ) ) .<_ ( N .\/ ( Y ./\ W ) ) )
19 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( X e. B /\ Y e. B ) /\ ( ( Q .\/ ( X ./\ W ) ) = X /\ ( N .\/ ( Y ./\ W ) ) = Y /\ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` N ) .(+) ( I ` ( Y ./\ W ) ) ) ) ) -> ( Q .\/ ( X ./\ W ) ) = X )
20 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( X e. B /\ Y e. B ) /\ ( ( Q .\/ ( X ./\ W ) ) = X /\ ( N .\/ ( Y ./\ W ) ) = Y /\ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` N ) .(+) ( I ` ( Y ./\ W ) ) ) ) ) -> ( N .\/ ( Y ./\ W ) ) = Y )
21 18 19 20 3brtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( N e. A /\ -. N .<_ W ) ) /\ ( X e. B /\ Y e. B ) /\ ( ( Q .\/ ( X ./\ W ) ) = X /\ ( N .\/ ( Y ./\ W ) ) = Y /\ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` N ) .(+) ( I ` ( Y ./\ W ) ) ) ) ) -> X .<_ Y )