Theorem cdlemk23-3 31384

Description: Part of proof of Lemma K of [Crawley] p. 118. Eliminate the ( R ` C ) =/= ( R ` D ) requirement from cdlemk22-3 31383. (Contributed by NM, 7-Jul-2013.)

Hypotheses

Ref	Expression
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

Ref

Expression

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 ) )

Distinct variable groups:   e, d, f, i, ./\   .<_ , i   .\/ , d, e, f, i   A, i   j, d, D, e, f, i   f, F, i   G, d, e, j   i, H   i, K   f, N, i   P, d, e, f, i   R, d, e, f, i   T, d, e, f, i   W, d, e, f, i   ./\ , j   .<_ , j   .\/ , j   A, j   j, F   j, H   j, K   j, N   P, j   R, j   S, d, e, j   T, j   j, W   F, d, e   .<_ , e   C, d, e, f, i, j   f, G, i   x, d, e, f, i, j

Allowed substitution hints:   A( x, e, f, d)   B( x, e, f, i, j, d)   C( x)   D( x)   P( x)   R( x)   S( x, f, i)   T( x)   F( x)   G( x)   H( x, e, f, d)   .\/ ( x)   K( x, e, f, d)   .<_ ( x, f, d)   ./\ ( x)   N( x, e, d)   W( x)   Y( x, e, f, i, j, d)

Proof of Theorem cdlemk23-3

Step	Hyp	Ref	Expression
1		simp11 987	. . 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 ) ) ) ) -> ( K e. HL /\ W e. H ) )
2		simp121 1089	. . 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 ) ) ) ) -> F e. T )
3		simp122 1090	. . 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 e. T )
4		simp123 1091	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> N e. T )
5		simp131 1092	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> G e. T )
6		simp133 1094	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> x e. T )
7	4, 5, 6	3jca 1134	. . 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 ) ) ) ) -> ( N e. T /\ G e. T /\ x e. T ) )
8		simp21 990	. . 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 ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
9		simp221 1098	. . 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 ) ) ) ) -> ( R ` F ) = ( R ` N ) )
10		simp222 1099	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> F =/= ( _I |` B ) )
11		simp223 1100	. . . 4 |- ( ( ( ( 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 =/= ( _I |` B ) )
12		simp231 1101	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> G =/= ( _I |` B ) )
13	10, 11, 12	3jca 1134	. . 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 ) ) ) ) -> ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
14		simp233 1103	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> x =/= ( _I |` B ) )
15		simp333 1112	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` G ) =/= ( R ` x ) )
16		simp332 1111	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` x ) =/= ( R ` F ) )
17	14, 15, 16	3jca 1134	. . 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 ) ) ) ) -> ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) )
18		simp313 1106	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` D ) =/= ( R ` F ) )
19		simp32l 1082	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` G ) =/= ( R ` D ) )
20		simp331 1110	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` x ) =/= ( R ` D ) )
21	18, 19, 20	3jca 1134	. . 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 ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` D ) ) )
22		cdlemk3.b	. . . 4 |- B = ( Base ` K )
23		cdlemk3.l	. . . 4 |- .<_ = ( le ` K )
24		cdlemk3.j	. . . 4 |- .\/ = ( join ` K )
25		cdlemk3.m	. . . 4 |- ./\ = ( meet ` K )
26		cdlemk3.a	. . . 4 |- A = ( Atoms ` K )
27		cdlemk3.h	. . . 4 |- H = ( LHyp ` K )
28		cdlemk3.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
29		cdlemk3.r	. . . 4 |- R = ( ( trL ` K ) ` W )
30		cdlemk3.s	. . . 4 |- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) )
31		cdlemk3.u1	. . . 4 |- Y = ( d e. T , e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( ( S ` d ) ` P ) .\/ ( R ` ( e o. `' d ) ) ) ) ) )
32	22, 23, 24, 25, 26, 27, 28, 29, 30, 31	cdlemk22-3 31383	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T /\ x e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` x ) =/= ( R ` D ) ) ) ) -> ( ( D Y G ) ` P ) = ( ( x Y G ) ` P ) )
33	1, 2, 3, 7, 8, 9, 13, 17, 21, 32	syl333anc 1216	. 2 |- ( ( ( ( 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 ) = ( ( x Y G ) ` P ) )
34		simp132 1093	. . 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 ) ) ) ) -> C e. T )
35		simp232 1102	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> C =/= ( _I |` B ) )
36	10, 35, 12	3jca 1134	. . 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 ) ) ) ) -> ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) )
37		simp312 1105	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` C ) =/= ( R ` F ) )
38		simp311 1104	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` G ) =/= ( R ` C ) )
39		simp32r 1083	. . . 4 |- ( ( ( ( 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 ) ) ) ) -> ( R ` x ) =/= ( R ` C ) )
40	37, 38, 39	3jca 1134	. . 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 ) ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` x ) =/= ( R ` C ) ) )
41	22, 23, 24, 25, 26, 27, 28, 29, 30, 31	cdlemk22-3 31383	. . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ G e. T /\ x e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( x =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` x ) /\ ( R ` x ) =/= ( R ` F ) ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` x ) =/= ( R ` C ) ) ) ) -> ( ( C Y G ) ` P ) = ( ( x Y G ) ` P ) )
42	1, 2, 34, 7, 8, 9, 36, 17, 40, 41	syl333anc 1216	. 2 |- ( ( ( ( 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 ) ) ) ) -> ( ( C Y G ) ` P ) = ( ( x Y G ) ` P ) )
43	33, 42	eqtr4d 2439	1 |- ( ( ( ( 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 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   class class class wbr 4172   e. cmpt 4226   _I cid 4453   `'ccnv 4836   |` cres 4839   o. ccom 4841   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   iota_crio 6501   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640

This theorem is referenced by:  cdlemk24-3  31385

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641