cdlemk24-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the ( Rx ) =/= ( RC ) requirement from cdlemk23-3 using ( RC ) = ( RD ) . (Contributed by NM, 7-Jul-2013)

	Ref	Expression
Hypotheses	cdlemk3.b	\|- B = ( Base ` K )
	cdlemk3.l	\|- .<_ = ( le ` K )
	cdlemk3.j	\|- .\/ = ( join ` K )
	cdlemk3.m	\|- ./\ = ( meet ` K )
	cdlemk3.a	\|- A = ( Atoms ` K )
	cdlemk3.h	\|- H = ( LHyp ` K )
	cdlemk3.t	\|- T = ( ( LTrn ` K ) ` W )
	cdlemk3.r	\|- R = ( ( trL ` K ) ` W )
	cdlemk3.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
	cdlemk3.u1	\|- Y = ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
Assertion	cdlemk24-3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk3.b	\|- B = ( Base ` K )
2		cdlemk3.l	\|- .<_ = ( le ` K )
3		cdlemk3.j	\|- .\/ = ( join ` K )
4		cdlemk3.m	\|- ./\ = ( meet ` K )
5		cdlemk3.a	\|- A = ( Atoms ` K )
6		cdlemk3.h	\|- H = ( LHyp ` K )
7		cdlemk3.t	\|- T = ( ( LTrn ` K ) ` W )
8		cdlemk3.r	\|- R = ( ( trL ` K ) ` W )
9		cdlemk3.s	\|- S = ( f e. T \|-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
10		cdlemk3.u1	\|- Y = ( d e. T , e e. T \|-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
11		simp31	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
12		simp32l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
13		simp331	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` x ) =/= ( R ` D ) )
14		simp32r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` C ) = ( R ` D ) )
15	13 14	neeqtrrd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` x ) =/= ( R ` C ) )
16	12 15	jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` C ) ) )
17		simp33	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) )
18	11 16 17	3jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) )
19	1 2 3 4 5 6 7 8 9 10	cdlemk23-3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` C ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
20	18 19	syld3an3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I \|` B ) /\ D =/= ( _I \|` B ) ) /\ ( G =/= ( _I \|` B ) /\ C =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) ) /\ ( ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) /\ ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )

Metamath Proof Explorer

Theorem cdlemk24-3

Proof