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Metamath Proof Explorer

Theorem 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 ) )