Metamath Proof Explorer


Theorem cdlemk25-3

Description: Part of proof of Lemma K of Crawley p. 118. Eliminate the ( RC ) = ( RD ) requirement from cdlemk24-3 . (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 cdlemk25-3
|- ( ( ( ( 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 ` 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 simpl1
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( G e. T /\ C e. T /\ x e. T ) ) )
12 simpl2
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( ( R ` F ) = ( R ` N ) /\ F =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( G =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ x =/= ( _I |` B ) ) ) )
13 simpl31
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) )
14 simpl32
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( R ` G ) =/= ( R ` D ) )
15 simpr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( R ` C ) = ( R ` D ) )
16 14 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) = ( R ` D ) ) )
17 simpl33
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( R ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) )
18 1 2 3 4 5 6 7 8 9 10 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 ) )
19 11 12 13 16 17 18 syl113anc
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) = ( R ` D ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
20 simp11
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( K e. HL /\ W e. H ) )
21 simp121
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F e. T )
22 simp122
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D e. T )
23 20 21 22 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
24 23 adantr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) )
25 simp123
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> N e. T )
26 simp131
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G e. T )
27 simp132
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C e. T )
28 25 26 27 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( N e. T /\ G e. T /\ C e. T ) )
29 simp21
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
30 simp221
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` F ) = ( R ` N ) )
31 28 29 30 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
32 31 adantr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) )
33 simp222
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> F =/= ( _I |` B ) )
34 simp223
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> D =/= ( _I |` B ) )
35 simp231
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> G =/= ( _I |` B ) )
36 33 34 35 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
37 36 adantr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
38 simp232
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> C =/= ( _I |` B ) )
39 simp311
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` G ) =/= ( R ` C ) )
40 simp312
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
41 38 39 40 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) )
42 41 adantr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) )
43 simp313
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
44 43 adantr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` D ) =/= ( R ` F ) )
45 simpl32
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` G ) =/= ( R ` D ) )
46 simpr
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( R ` C ) =/= ( R ` D ) )
47 44 45 46 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 ` x ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) )
48 1 2 3 4 5 6 7 8 9 10 cdlemk22-3
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ C e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( C =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` C ) =/= ( R ` F ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` C ) =/= ( R ` D ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
49 24 32 37 42 47 48 syl113anc
 |-  ( ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) /\ ( R ` C ) =/= ( R ` D ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )
50 19 49 pm2.61dane
 |-  ( ( ( ( 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 ` D ) /\ ( R ` x ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` x ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( C Y G ) ` P ) )