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Theorem cdlemg27b 30885
Description: TODO: Fix comment. (Contributed by NM, 28-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
cdlemg27b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  ( R `  F )  .<_  ( Q 
.\/  z ) )

Proof of Theorem cdlemg27b
StepHypRef Expression
1 simp11 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 986 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp22 989 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
5 simp23l 1076 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  F  e.  T
)
6 simp31 991 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  v  =/=  ( R `  F )
)
7 cdlemg12.l . . . . . 6  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . . . 6  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . . . 6  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . . . 6  |-  H  =  ( LHyp `  K
)
12 cdlemg12.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg12b.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
14 cdlemg31.n . . . . . 6  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
157, 8, 9, 10, 11, 12, 13, 14cdlemg31b0a 30884 . . . . 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  =  ( 0. `  K ) ) )
161, 2, 3, 4, 5, 6, 15syl132anc 1200 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K
) ) )
17 simp23r 1077 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  z  =/=  N
)
1817adantr 451 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  ( N  e.  A  \/  N  =  ( 0. `  K
) ) )  -> 
z  =/=  N )
19 simp11l 1066 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  K  e.  HL )
2019adantr 451 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  K  e.  HL )
21 hlatl 29550 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
2220, 21syl 15 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  K  e.  AtLat
)
23 simpl21 1033 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  z  e.  A )
24 simpr 447 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  N  e.  A )
257, 10atcmp 29501 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  z  e.  A  /\  N  e.  A )  ->  (
z  .<_  N  <->  z  =  N ) )
2622, 23, 24, 25syl3anc 1182 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  ( z  .<_  N  <->  z  =  N ) )
2726necon3bbid 2480 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  e.  A
)  ->  ( -.  z  .<_  N  <->  z  =/=  N ) )
2819adantr 451 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  K  e.  HL )
2928, 21syl 15 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  K  e.  AtLat
)
30 simpl21 1033 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  z  e.  A )
31 eqid 2283 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
327, 31, 10atnle0 29499 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  z  e.  A )  ->  -.  z  .<_  ( 0. `  K ) )
3329, 30, 32syl2anc 642 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  -.  z  .<_  ( 0. `  K
) )
34 simpr 447 . . . . . . . . 9  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  N  =  ( 0. `  K ) )
3534breq2d 4035 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  ( z  .<_  N  <->  z  .<_  ( 0.
`  K ) ) )
3633, 35mtbird 292 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  -.  z  .<_  N )
3717adantr 451 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  z  =/=  N )
3836, 372thd 231 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  N  =  ( 0. `  K ) )  ->  ( -.  z  .<_  N  <->  z  =/=  N ) )
3927, 38jaodan 760 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  ( N  e.  A  \/  N  =  ( 0. `  K
) ) )  -> 
( -.  z  .<_  N 
<->  z  =/=  N ) )
4018, 39mpbird 223 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  /\  ( N  e.  A  \/  N  =  ( 0. `  K
) ) )  ->  -.  z  .<_  N )
4116, 40mpdan 649 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  z  .<_  N )
42 simp32 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  z  .<_  ( P 
.\/  v ) )
43 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4419, 43syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  K  e.  Lat )
45 simp21 988 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  z  e.  A
)
46 eqid 2283 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
4746, 10atbase 29479 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  ( Base `  K
) )
4845, 47syl 15 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  z  e.  (
Base `  K )
)
49 simp12l 1068 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  P  e.  A
)
50 simp22l 1074 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  v  e.  A
)
5146, 8, 10hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  ( P  .\/  v
)  e.  ( Base `  K ) )
5219, 49, 50, 51syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( P  .\/  v )  e.  (
Base `  K )
)
53 simp13l 1070 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  Q  e.  A
)
54 simp33 993 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( F `  P )  =/=  P
)
557, 10, 11, 12, 13trlat 30358 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
561, 2, 5, 54, 55syl112anc 1186 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
5746, 8, 10hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( R `  F )  e.  A )  -> 
( Q  .\/  ( R `  F )
)  e.  ( Base `  K ) )
5819, 53, 56, 57syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( Q  .\/  ( R `  F ) )  e.  ( Base `  K ) )
5946, 7, 9latlem12 14184 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( z  e.  (
Base `  K )  /\  ( P  .\/  v
)  e.  ( Base `  K )  /\  ( Q  .\/  ( R `  F ) )  e.  ( Base `  K
) ) )  -> 
( ( z  .<_  ( P  .\/  v )  /\  z  .<_  ( Q 
.\/  ( R `  F ) ) )  <-> 
z  .<_  ( ( P 
.\/  v )  ./\  ( Q  .\/  ( R `
 F ) ) ) ) )
6044, 48, 52, 58, 59syl13anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( ( z 
.<_  ( P  .\/  v
)  /\  z  .<_  ( Q  .\/  ( R `
 F ) ) )  <->  z  .<_  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) ) ) )
6114breq2i 4031 . . . . . 6  |-  ( z 
.<_  N  <->  z  .<_  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) ) )
6260, 61syl6bbr 254 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( ( z 
.<_  ( P  .\/  v
)  /\  z  .<_  ( Q  .\/  ( R `
 F ) ) )  <->  z  .<_  N ) )
6362biimpd 198 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( ( z 
.<_  ( P  .\/  v
)  /\  z  .<_  ( Q  .\/  ( R `
 F ) ) )  ->  z  .<_  N ) )
6442, 63mpand 656 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( z  .<_  ( Q  .\/  ( R `
 F ) )  ->  z  .<_  N ) )
6541, 64mtod 168 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  z  .<_  ( Q  .\/  ( R `
 F ) ) )
667, 11, 12, 13trlle 30373 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
671, 5, 66syl2anc 642 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  .<_  W )
68 simp13r 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  Q  .<_  W )
69 nbrne2 4041 . . . 4  |-  ( ( ( R `  F
)  .<_  W  /\  -.  Q  .<_  W )  -> 
( R `  F
)  =/=  Q )
7067, 68, 69syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  =/=  Q
)
717, 8, 10hlatexch1 29584 . . 3  |-  ( ( K  e.  HL  /\  ( ( R `  F )  e.  A  /\  z  e.  A  /\  Q  e.  A
)  /\  ( R `  F )  =/=  Q
)  ->  ( ( R `  F )  .<_  ( Q  .\/  z
)  ->  z  .<_  ( Q  .\/  ( R `
 F ) ) ) )
7219, 56, 45, 53, 70, 71syl131anc 1195 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  ( ( R `
 F )  .<_  ( Q  .\/  z )  ->  z  .<_  ( Q 
.\/  ( R `  F ) ) ) )
7365, 72mtod 168 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( z  e.  A  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  z  =/= 
N ) )  /\  ( v  =/=  ( R `  F )  /\  z  .<_  ( P 
.\/  v )  /\  ( F `  P )  =/=  P ) )  ->  -.  ( R `  F )  .<_  ( Q 
.\/  z ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg28b  30892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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