cdlemkyuu

Description: cdlemkyu with some hypotheses eliminated. TODO: Clean all this up. (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 ) ) ) )
	cdlemk5c.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk5a.u2	\|- C = ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )
Assertion	cdlemkyuu	\|- ( ( ( ( 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 ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) )

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		cdlemk5c.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
12		cdlemk5a.u2	\|- C = ( e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` b ) ` P ) .\/ ( R ` ( e o. `' b ) ) ) ) ) )
13		eqid	\|- ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) ) = ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
14		eqid	\|- ( S ` b ) = ( S ` b )
15	1 2 3 4 5 6 7 8 9 10 11 13 14 12	cdlemkyu	\|- ( ( ( ( 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 ) ) ) ) -> [_ G / g ]_ Y = ( ( C ` G ) ` P ) )

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