Theorem cdlemk11tb

Description: Part of proof of Lemma K of Crawley p. 118. Lemma for Eq. 5, p. 119. G , I stand for g, h. cdlemk11ta with hypotheses removed. TODO: Can this be proved directly with no quantification? (Contributed by NM, 21-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 ) ) ) )
Assertion	cdlemk11tb	|- ( ( ( ( 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 ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ G / g ]_ Y .<_ ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) )

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		eqid	|- ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
12		eqid	|- ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) ) = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )
13	1 2 3 4 5 6 7 8 9 10 11 12	cdlemk11ta	|- ( ( ( ( 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 ) ) /\ ( b e. T /\ ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) /\ ( I e. T /\ I =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` I ) ) ) ) -> [_ G / g ]_ Y .<_ ( [_ I / g ]_ Y .\/ ( R ` ( I o. `' G ) ) ) )