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Theorem cdlemg27b 34436
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 1018 . . . . 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 1019 . . . . 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 1020 . . . . 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 1022 . . . . 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 1109 . . . . 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 1024 . . . . 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 34435 . . . . 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 1236 . . . 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 1110 . . . . . 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 465 . . . . 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 1099 . . . . . . . . . 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 465 . . . . . . . . 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 33101 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  AtLat )
2220, 21syl 16 . . . . . . . 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 1066 . . . . . . . 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 461 . . . . . . . 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 33052 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  z  e.  A  /\  N  e.  A )  ->  (
z  .<_  N  <->  z  =  N ) )
2622, 23, 24, 25syl3anc 1218 . . . . . . 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 2636 . . . . . 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 465 . . . . . . . . . 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 16 . . . . . . . . 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 1066 . . . . . . . . 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 2443 . . . . . . . . . 10  |-  ( 0.
`  K )  =  ( 0. `  K
)
327, 31, 10atnle0 33050 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  z  e.  A )  ->  -.  z  .<_  ( 0. `  K ) )
3329, 30, 32syl2anc 661 . . . . . . . 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 461 . . . . . . . . 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 4325 . . . . . . . 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 301 . . . . . . 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 465 . . . . . . 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 240 . . . . . 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 783 . . . . 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 232 . . . 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 668 . . 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 1025 . . . 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 33104 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
4419, 43syl 16 . . . . . . 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 1021 . . . . . . . 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 2443 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
4746, 10atbase 33030 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  ( Base `  K
) )
4845, 47syl 16 . . . . . . 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 1101 . . . . . . . 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 1107 . . . . . . . 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 33107 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  v  e.  A )  ->  ( P  .\/  v
)  e.  ( Base `  K ) )
5219, 49, 50, 51syl3anc 1218 . . . . . . 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 1103 . . . . . . . 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 1026 . . . . . . . . 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 33909 . . . . . . . . 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 1222 . . . . . . . 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 33107 . . . . . . . 8  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  ( R `  F )  e.  A )  -> 
( Q  .\/  ( R `  F )
)  e.  ( Base `  K ) )
5819, 53, 56, 57syl3anc 1218 . . . . . . 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 15269 . . . . . . 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 1220 . . . . . 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 4321 . . . . . 6  |-  ( z 
.<_  N  <->  z  .<_  ( ( P  .\/  v ) 
./\  ( Q  .\/  ( R `  F ) ) ) )
6260, 61syl6bbr 263 . . . . 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 207 . . . 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 675 . . 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 177 . 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 33924 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
671, 5, 66syl2anc 661 . . . 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 1104 . . . 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 4331 . . . 4  |-  ( ( ( R `  F
)  .<_  W  /\  -.  Q  .<_  W )  -> 
( R `  F
)  =/=  Q )
7067, 68, 69syl2anc 661 . . 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 33135 . . 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 1231 . 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 177 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   Basecbs 14195   lecple 14266   joincjn 15135   meetcmee 15136   0.cp0 15228   Latclat 15236   Atomscatm 33004   AtLatcal 33005   HLchlt 33091   LHypclh 33724   LTrncltrn 33841   trLctrl 33898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-map 7237  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899
This theorem is referenced by:  cdlemg28b  34443
  Copyright terms: Public domain W3C validator