Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg29 Unicode version

Theorem cdlemg29 31516
Description: Eliminate  ( F `
 P )  =/= 
P and  ( G `  P )  =/=  P from cdlemg28 31515. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
cdlemg33.o  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
Assertion
Ref Expression
cdlemg29  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Distinct variable groups:    z, A    z, F    z, H    z,  .\/    z, K    z,  .<_    z, N    z, P    z, Q    z, R    z, T    z, W    z, v    z, G   
z, O
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v)    T( v)    F( v)    G( v)    H( v)    .\/ ( v)    K( v)   
.<_ ( v)    ./\ ( z, v)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg29
StepHypRef Expression
1 simpl11 1030 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simpl12 1031 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
3 simpl13 1032 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp23l 1076 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  ->  F  e.  T )
54adantr 451 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  ->  F  e.  T )
6 simp23r 1077 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  ->  G  e.  T )
76adantr 451 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  ->  G  e.  T )
8 simpr 447 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  -> 
( F `  P
)  =  P )
9 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
10 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
11 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
12 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
13 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
14 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
15 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
169, 10, 11, 12, 13, 14, 15cdlemg14f 31464 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( F `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
171, 2, 3, 5, 7, 8, 16syl123anc 1199 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( F `  P )  =  P )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
18 simpl11 1030 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  -> 
( K  e.  HL  /\  W  e.  H ) )
19 simpl12 1031 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
20 simpl13 1032 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
214adantr 451 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  ->  F  e.  T )
226adantr 451 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  ->  G  e.  T )
23 simpr 447 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  -> 
( G `  P
)  =  P )
249, 10, 11, 12, 13, 14, 15cdlemg14g 31465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  ( G `  P )  =  P ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
2518, 19, 20, 21, 22, 23, 24syl123anc 1199 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( G `  P )  =  P )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
26 simpl1 958 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
27 simpl2 959 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( (
v  e.  A  /\  v  .<_  W )  /\  ( z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) ) )
28 simp31l 1078 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  -> 
z  =/=  N )
2928adantr 451 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  z  =/=  N )
30 simp31r 1079 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  -> 
z  =/=  O )
3130adantr 451 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  z  =/=  O )
32 simpl32 1037 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  z  .<_  ( P  .\/  v ) )
3329, 31, 323jca 1132 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( z  =/=  N  /\  z  =/= 
O  /\  z  .<_  ( P  .\/  v ) ) )
34 simpl33 1038 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) )
35 simpr 447 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( ( F `  P )  =/=  P  /\  ( G `
 P )  =/= 
P ) )
36 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
37 cdlemg33.o . . . 4  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
389, 10, 11, 12, 13, 14, 15, 36, 37cdlemg28 31515 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P )  =/=  P ) ) )  ->  ( ( P 
.\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
3926, 27, 33, 34, 35, 38syl113anc 1194 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  /\  ( ( F `  P )  =/=  P  /\  ( G `  P
)  =/=  P ) )  ->  ( ( P  .\/  ( F `  ( G `  P ) ) )  ./\  W
)  =  ( ( Q  .\/  ( F `
 ( G `  Q ) ) ) 
./\  W ) )
4017, 25, 39pm2.61da2ne 2538 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  (
z  e.  A  /\  -.  z  .<_  W )  /\  ( F  e.  T  /\  G  e.  T ) )  /\  ( ( z  =/= 
N  /\  z  =/=  O )  /\  z  .<_  ( P  .\/  v )  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  W )  =  ( ( Q 
.\/  ( F `  ( G `  Q ) ) )  ./\  W
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  cdlemg34  31523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970
  Copyright terms: Public domain W3C validator