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Theorem cdlemg33b0 34627
Description: TODO: Fix comment. (Contributed by NM, 30-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 ) ) )
Assertion
Ref Expression
cdlemg33b0  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  .<_  ( P  .\/  v ) ) ) )
Distinct variable groups:    A, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r   
z, A    z, F, r    H, r, z    z,  .\/    K, r, z    z,  .<_    N, r, z    z, P   
z, Q    z, R    z, T    z, W    z,
v, r
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v, r)    T( v, r)    F( v)    H( v)    .\/ ( v)    K( v)   
.<_ ( v)    ./\ ( z, v, r)    N( v)    W( v)

Proof of Theorem cdlemg33b0
StepHypRef Expression
1 simp11 1018 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 1019 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 1020 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp22 1022 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  e.  A
)
5 simp21l 1105 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  e.  A
)
6 simp21r 1106 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  .<_  W )
75, 6jca 532 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
8 simp23 1023 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F  e.  T
)
9 simp32 1025 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  =/=  ( R `  F )
)
10 cdlemg12.l . . . . . 6  |-  .<_  =  ( le `  K )
11 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
12 cdlemg12.m . . . . . 6  |-  ./\  =  ( meet `  K )
13 cdlemg12.a . . . . . 6  |-  A  =  ( Atoms `  K )
14 cdlemg12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 cdlemg12.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
16 cdlemg12b.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
17 cdlemg31.n . . . . . 6  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
1810, 11, 12, 13, 14, 15, 16, 17cdlemg31d 34626 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )  /\  N  e.  A
) )  ->  -.  N  .<_  W )
191, 2, 3, 7, 8, 9, 4, 18syl133anc 1242 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  -.  N  .<_  W )
204, 19jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( N  e.  A  /\  -.  N  .<_  W ) )
21 simp31 1024 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  =/=  Q
)
22 nbrne2 4394 . . . . . 6  |-  ( ( v  .<_  W  /\  -.  N  .<_  W )  ->  v  =/=  N
)
2322necomd 2716 . . . . 5  |-  ( ( v  .<_  W  /\  -.  N  .<_  W )  ->  N  =/=  v
)
246, 19, 23syl2anc 661 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  =/=  v
)
255, 24jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( v  e.  A  /\  N  =/=  v ) )
26 simp33 1026 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
2710, 11, 13, 144atex3 34007 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( N  e.  A  /\  -.  N  .<_  W ) )  /\  ( P  =/=  Q  /\  ( v  e.  A  /\  N  =/=  v
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  v  /\  z  .<_  ( N 
.\/  v ) ) ) )
281, 2, 3, 20, 21, 25, 26, 27syl133anc 1242 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  v  /\  z  .<_  ( N 
.\/  v ) ) ) )
29 df-3an 967 . . . . 5  |-  ( ( z  =/=  N  /\  z  =/=  v  /\  z  .<_  ( N  .\/  v
) )  <->  ( (
z  =/=  N  /\  z  =/=  v )  /\  z  .<_  ( N  .\/  v ) ) )
30 simpl 457 . . . . . . 7  |-  ( ( z  =/=  N  /\  z  =/=  v )  -> 
z  =/=  N )
3130a1i 11 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( (
z  =/=  N  /\  z  =/=  v )  -> 
z  =/=  N ) )
32 simp12l 1101 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  P  e.  A
)
33 simp13l 1103 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  Q  e.  A
)
3410, 11, 12, 13, 14, 15, 16, 17cdlemg31a 34623 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( P  .\/  v ) )
351, 32, 33, 5, 8, 34syl122anc 1228 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  .<_  ( P 
.\/  v ) )
36 simp11l 1099 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  HL )
3710, 11, 13hlatlej2 33302 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  v  .<_  ( P  .\/  v ) )
3836, 32, 5, 37syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  .<_  ( P 
.\/  v ) )
39 hllat 33290 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
4036, 39syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  K  e.  Lat )
41 eqid 2450 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
4241, 13atbase 33216 . . . . . . . . . . 11  |-  ( N  e.  A  ->  N  e.  ( Base `  K
) )
434, 42syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  N  e.  (
Base `  K )
)
4441, 13atbase 33216 . . . . . . . . . . 11  |-  ( v  e.  A  ->  v  e.  ( Base `  K
) )
455, 44syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  e.  (
Base `  K )
)
4641, 11, 13hlatjcl 33293 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  ( P  .\/  v
)  e.  ( Base `  K ) )
4736, 32, 5, 46syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  .\/  v )  e.  (
Base `  K )
)
4841, 10, 11latjle12 15320 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( N  e.  ( Base `  K )  /\  v  e.  ( Base `  K )  /\  ( P  .\/  v )  e.  ( Base `  K
) ) )  -> 
( ( N  .<_  ( P  .\/  v )  /\  v  .<_  ( P 
.\/  v ) )  <-> 
( N  .\/  v
)  .<_  ( P  .\/  v ) ) )
4940, 43, 45, 47, 48syl13anc 1221 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( N 
.<_  ( P  .\/  v
)  /\  v  .<_  ( P  .\/  v ) )  <->  ( N  .\/  v )  .<_  ( P 
.\/  v ) ) )
5035, 38, 49mpbi2and 912 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( N  .\/  v )  .<_  ( P 
.\/  v ) )
5150adantr 465 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( N  .\/  v )  .<_  ( P 
.\/  v ) )
5240adantr 465 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  K  e.  Lat )
5341, 13atbase 33216 . . . . . . . . 9  |-  ( z  e.  A  ->  z  e.  ( Base `  K
) )
5453adantl 466 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  z  e.  ( Base `  K )
)
5541, 11, 13hlatjcl 33293 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  N  e.  A  /\  v  e.  A )  ->  ( N  .\/  v
)  e.  ( Base `  K ) )
5636, 4, 5, 55syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( N  .\/  v )  e.  (
Base `  K )
)
5756adantr 465 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( N  .\/  v )  e.  (
Base `  K )
)
5847adantr 465 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( P  .\/  v )  e.  (
Base `  K )
)
5941, 10lattr 15314 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( z  e.  (
Base `  K )  /\  ( N  .\/  v
)  e.  ( Base `  K )  /\  ( P  .\/  v )  e.  ( Base `  K
) ) )  -> 
( ( z  .<_  ( N  .\/  v )  /\  ( N  .\/  v )  .<_  ( P 
.\/  v ) )  ->  z  .<_  ( P 
.\/  v ) ) )
6052, 54, 57, 58, 59syl13anc 1221 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( (
z  .<_  ( N  .\/  v )  /\  ( N  .\/  v )  .<_  ( P  .\/  v ) )  ->  z  .<_  ( P  .\/  v ) ) )
6151, 60mpan2d 674 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( z  .<_  ( N  .\/  v
)  ->  z  .<_  ( P  .\/  v ) ) )
6231, 61anim12d 563 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( (
( z  =/=  N  /\  z  =/=  v
)  /\  z  .<_  ( N  .\/  v ) )  ->  ( z  =/=  N  /\  z  .<_  ( P  .\/  v ) ) ) )
6329, 62syl5bi 217 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( (
z  =/=  N  /\  z  =/=  v  /\  z  .<_  ( N  .\/  v
) )  ->  (
z  =/=  N  /\  z  .<_  ( P  .\/  v ) ) ) )
6463anim2d 565 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  /\  z  e.  A
)  ->  ( ( -.  z  .<_  W  /\  ( z  =/=  N  /\  z  =/=  v  /\  z  .<_  ( N 
.\/  v ) ) )  ->  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  .<_  ( P  .\/  v ) ) ) ) )
6564reximdva 2910 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  v  /\  z  .<_  ( N  .\/  v
) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  .<_  ( P  .\/  v ) ) ) ) )
6628, 65mpd 15 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  N  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  .<_  ( P  .\/  v ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757    =/= wne 2641   E.wrex 2793   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   lecple 14333   joincjn 15202   meetcmee 15203   Latclat 15303   Atomscatm 33190   HLchlt 33277   LHypclh 33910   LTrncltrn 34027   trLctrl 34084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-map 7302  df-poset 15204  df-plt 15216  df-lub 15232  df-glb 15233  df-join 15234  df-meet 15235  df-p0 15297  df-p1 15298  df-lat 15304  df-clat 15366  df-oposet 33103  df-ol 33105  df-oml 33106  df-covers 33193  df-ats 33194  df-atl 33225  df-cvlat 33249  df-hlat 33278  df-llines 33424  df-lplanes 33425  df-psubsp 33429  df-pmap 33430  df-padd 33722  df-lhyp 33914  df-laut 33915  df-ldil 34030  df-ltrn 34031  df-trl 34085
This theorem is referenced by:  cdlemg33b  34633  cdlemg33c  34634
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