Metamath Proof Explorer

Theorem cdlemk39

Description: Part of proof of Lemma K of Crawley p. 118. Line 31, p. 119. Trace-preserving property of tau, represented by X . (Contributed by NM, 19-Jul-2013)

Ref Expression
Hypotheses cdlemk4.b 
|- B = ( Base ` K )
cdlemk4.l 
|- .<_ = ( le ` K )
cdlemk4.j 
|- .\/ = ( join ` K )
cdlemk4.m 
|- ./\ = ( meet ` K )
cdlemk4.a 
|- A = ( Atoms ` K )
cdlemk4.h 
|- H = ( LHyp ` K )
cdlemk4.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk4.r 
|- R = ( ( trL ` K ) ` W )
cdlemk4.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk4.y 
|- Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) )
cdlemk4.x 
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) )
Assertion cdlemk39 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` G ) )

Proof

Step Hyp Ref Expression
1 cdlemk4.b 
 |-  B = ( Base ` K )
2 cdlemk4.l 
 |-  .<_ = ( le ` K )
3 cdlemk4.j 
 |-  .\/ = ( join ` K )
4 cdlemk4.m 
 |-  ./\ = ( meet ` K )
5 cdlemk4.a 
 |-  A = ( Atoms ` K )
6 cdlemk4.h 
 |-  H = ( LHyp ` K )
7 cdlemk4.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk4.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk4.z 
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk4.y 
 |-  Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) )
11 cdlemk4.x 
 |-  X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) )
12 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> K e. HL )
13 simp3ll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P e. A )
14 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G e. T )
16 simp22r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G =/= ( _I |` B ) )
17 1 5 6 7 8 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
18 14 15 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` G ) e. A )
19 2 3 5 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> P .<_ ( P .\/ ( R ` G ) ) )
20 12 13 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P .<_ ( P .\/ ( R ` G ) ) )
21 1 2 3 4 5 6 7 8 9 10 11 cdlemk38 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) .<_ ( P .\/ ( R ` G ) ) )
22 12 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> K e. Lat )
23 1 5 atbase 
 |-  ( P e. A -> P e. B )
24 13 23 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P e. B )
25 1 2 3 4 5 6 7 8 9 10 11 cdlemk35 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> X e. T )
26 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A )
27 14 25 13 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) e. A )
28 1 5 atbase 
 |-  ( ( X ` P ) e. A -> ( X ` P ) e. B )
29 27 28 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) e. B )
30 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B )
31 12 13 18 30 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( R ` G ) ) e. B )
32 1 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( P e. B /\ ( X ` P ) e. B /\ ( P .\/ ( R ` G ) ) e. B ) ) -> ( ( P .<_ ( P .\/ ( R ` G ) ) /\ ( X ` P ) .<_ ( P .\/ ( R ` G ) ) ) <-> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) ) )
33 22 24 29 31 32 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .<_ ( P .\/ ( R ` G ) ) /\ ( X ` P ) .<_ ( P .\/ ( R ` G ) ) ) <-> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) ) )
34 20 21 33 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) )
35 1 3 5 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ ( X ` P ) e. A ) -> ( P .\/ ( X ` P ) ) e. B )
36 12 13 27 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( X ` P ) ) e. B )
37 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> W e. H )
38 1 6 lhpbase 
 |-  ( W e. H -> W e. B )
39 37 38 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> W e. B )
40 1 2 4 latmlem1 
 |-  ( ( K e. Lat /\ ( ( P .\/ ( X ` P ) ) e. B /\ ( P .\/ ( R ` G ) ) e. B /\ W e. B ) ) -> ( ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) ) )
41 22 36 31 39 40 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) ) )
42 34 41 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) )
43 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P e. A /\ -. P .<_ W ) )
44 2 3 4 5 6 7 8 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` X ) = ( ( P .\/ ( X ` P ) ) ./\ W ) )
45 14 25 43 44 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) = ( ( P .\/ ( X ` P ) ) ./\ W ) )
46 2 3 4 5 6 7 8 trlval5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( R ` G ) ) ./\ W ) )
47 14 15 43 46 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` G ) = ( ( P .\/ ( R ` G ) ) ./\ W ) )
48 42 45 47 3brtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` G ) )