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Metamath Proof Explorer

Theorem cdlemk48

Description: Part of proof of Lemma K of Crawley p. 118. Line 4, p. 120. G , I stand for g, h. X represents tau. (Contributed by NM, 22-Jul-2013)

Ref Expression
Hypotheses cdlemk5.b 
|- B = ( Base ` K )
cdlemk5.l 
|- .<_ = ( le ` K )
cdlemk5.j 
|- .\/ = ( join ` K )
cdlemk5.m 
|- ./\ = ( meet ` K )
cdlemk5.a 
|- A = ( Atoms ` K )
cdlemk5.h 
|- H = ( LHyp ` K )
cdlemk5.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk5.r 
|- R = ( ( trL ` K ) ` W )
cdlemk5.z 
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
cdlemk5.y 
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdlemk5.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 cdlemk48 
|- ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )

Proof

Step Hyp Ref Expression
1 cdlemk5.b 
 |-  B = ( Base ` K )
2 cdlemk5.l 
 |-  .<_ = ( le ` K )
3 cdlemk5.j 
 |-  .\/ = ( join ` K )
4 cdlemk5.m 
 |-  ./\ = ( meet ` K )
5 cdlemk5.a 
 |-  A = ( Atoms ` K )
6 cdlemk5.h 
 |-  H = ( LHyp ` K )
7 cdlemk5.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemk5.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemk5.z 
 |-  Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) )
10 cdlemk5.y 
 |-  Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
11 cdlemk5.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 simp11l 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> K e. HL )
13 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> K e. Lat )
14 simp11 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( K e. HL /\ W e. H ) )
15 simp12 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( F e. T /\ F =/= ( _I |` B ) ) )
16 simp13 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( G e. T /\ G =/= ( _I |` B ) ) )
17 simp21 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> N e. T )
18 simp22 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( P e. A /\ -. P .<_ W ) )
19 simp23 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( R ` F ) = ( R ` N ) )
20 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s 
 |-  ( ( ( 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 / g ]_ X e. T )
21 14 15 16 17 18 19 20 syl132anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> [_ G / g ]_ X e. T )
22 simp3 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( I e. T /\ I =/= ( _I |` B ) ) )
23 1 2 3 4 5 6 7 8 9 10 11 cdlemk35s 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> [_ I / g ]_ X e. T )
24 14 15 22 17 18 19 23 syl132anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> [_ I / g ]_ X e. T )
25 6 7 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T /\ [_ I / g ]_ X e. T ) -> ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T )
26 14 21 24 25 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T )
27 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> P e. A )
28 2 5 6 7 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( [_ G / g ]_ X o. [_ I / g ]_ X ) e. T /\ P e. A ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A )
29 14 26 27 28 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A )
30 1 5 atbase 
 |-  ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. A -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. B )
31 29 30 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. B )
32 1 6 7 8 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ [_ G / g ]_ X e. T ) -> ( R ` [_ G / g ]_ X ) e. B )
33 14 21 32 syl2anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( R ` [_ G / g ]_ X ) e. B )
34 1 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) e. B /\ ( R ` [_ G / g ]_ X ) e. B ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )
35 13 31 33 34 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )
36 2 3 5 6 7 8 trlcoabs 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( [_ G / g ]_ X e. T /\ [_ I / g ]_ X e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .\/ ( R ` [_ G / g ]_ X ) ) = ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )
37 14 21 24 18 36 syl121anc 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .\/ ( R ` [_ G / g ]_ X ) ) = ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )
38 35 37 breqtrd 
 |-  ( ( ( ( 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 ) ) /\ ( I e. T /\ I =/= ( _I |` B ) ) ) -> ( ( [_ G / g ]_ X o. [_ I / g ]_ X ) ` P ) .<_ ( ( [_ I / g ]_ X ` P ) .\/ ( R ` [_ G / g ]_ X ) ) )