cdlemk26b-3

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 14-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	cdlemk26b-3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. x e. T ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) )

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		simpl1	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
12	1 6 7 8	cdlemftr2	\|- ( ( K e. HL /\ W e. H ) -> E. x e. T ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) )
13	11 12	syl	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. x e. T ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) )
14		simp3r	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) )
15		simp11	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
16		simp133	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( R ` F ) = ( R ` N ) )
17		simp131	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> G e. T )
18		simp121	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> F e. T )
19		simp3l	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> x e. T )
20		simp123	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> N e. T )
21		simp3r2	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( R ` x ) =/= ( R ` F ) )
22		simp3r3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( R ` x ) =/= ( R ` G ) )
23	21 22	jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) )
24		simp122	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> F =/= ( _I \|` B ) )
25		simp132	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> G =/= ( _I \|` B ) )
26		simp3r1	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> x =/= ( _I \|` B ) )
27	24 25 26	3jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( F =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) )
28		simp2	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
29	1 2 3 4 5 6 7 8 9 10	cdlemkuel-3	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ x e. T /\ N e. T ) /\ ( ( ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( F =/= ( _I \|` B ) /\ G =/= ( _I \|` B ) /\ x =/= ( _I \|` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( x Y G ) e. T )
30	15 16 17 18 19 20 23 27 28 29	syl333anc	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( x Y G ) e. T )
31	14 30	jca	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) /\ ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) ) -> ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) )
32	31	3expia	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( x e. T /\ ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) ) -> ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) ) )
33	32	expd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( x e. T -> ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) -> ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) ) ) )
34	33	reximdvai	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( E. x e. T ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) -> E. x e. T ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) ) )
35	13 34	mpd	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ F =/= ( _I \|` B ) /\ N e. T ) /\ ( G e. T /\ G =/= ( _I \|` B ) /\ ( R ` F ) = ( R ` N ) ) ) /\ ( P e. A /\ -. P .<_ W ) ) -> E. x e. T ( ( x =/= ( _I \|` B ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` x ) =/= ( R ` G ) ) /\ ( x Y G ) e. T ) )

Metamath Proof Explorer

Theorem cdlemk26b-3

Proof