Metamath Proof Explorer

Theorem dihord2cN

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihjust.b 
|- B = ( Base ` K )
dihjust.l 
|- .<_ = ( le ` K )
dihjust.j 
|- .\/ = ( join ` K )
dihjust.m 
|- ./\ = ( meet ` K )
dihjust.a 
|- A = ( Atoms ` K )
dihjust.h 
|- H = ( LHyp ` K )
dihjust.i 
|- I = ( ( DIsoB ` K ) ` W )
dihjust.J 
|- J = ( ( DIsoC ` K ) ` W )
dihjust.u 
|- U = ( ( DVecH ` K ) ` W )
dihjust.s 
|- .(+) = ( LSSum ` U )
dihord2c.t 
|- T = ( ( LTrn ` K ) ` W )
dihord2c.r 
|- R = ( ( trL ` K ) ` W )
dihord2c.o 
|- O = ( h e. T |-> ( _I |` B ) )
Assertion dihord2cN 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> <. f , O >. e. ( I ` ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 dihjust.b 
 |-  B = ( Base ` K )
2 dihjust.l 
 |-  .<_ = ( le ` K )
3 dihjust.j 
 |-  .\/ = ( join ` K )
4 dihjust.m 
 |-  ./\ = ( meet ` K )
5 dihjust.a 
 |-  A = ( Atoms ` K )
6 dihjust.h 
 |-  H = ( LHyp ` K )
7 dihjust.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 dihjust.J 
 |-  J = ( ( DIsoC ` K ) ` W )
9 dihjust.u 
 |-  U = ( ( DVecH ` K ) ` W )
10 dihjust.s 
 |-  .(+) = ( LSSum ` U )
11 dihord2c.t 
 |-  T = ( ( LTrn ` K ) ` W )
12 dihord2c.r 
 |-  R = ( ( trL ` K ) ` W )
13 dihord2c.o 
 |-  O = ( h e. T |-> ( _I |` B ) )
14 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) )
15 eqidd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> O = O )
16 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> ( K e. HL /\ W e. H ) )
17 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> K e. HL )
18 17 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> K e. Lat )
19 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> X e. B )
20 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> W e. H )
21 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
22 20 21 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> W e. B )
23 1 4 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B )
24 18 19 22 23 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> ( X ./\ W ) e. B )
25 1 2 4 latmle2 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W )
26 18 19 22 25 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> ( X ./\ W ) .<_ W )
27 1 2 6 11 12 13 7 dibopelval3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( X ./\ W ) e. B /\ ( X ./\ W ) .<_ W ) ) -> ( <. f , O >. e. ( I ` ( X ./\ W ) ) <-> ( ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ O = O ) ) )
28 16 24 26 27 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> ( <. f , O >. e. ( I ` ( X ./\ W ) ) <-> ( ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ O = O ) ) )
29 14 15 28 mpbir2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) -> <. f , O >. e. ( I ` ( X ./\ W ) ) )