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Theorem 2lplnj 30431
Description: The join of two different lattice planes in a (3-dimensional) lattice volume equals the volume. (Contributed by NM, 12-Jul-2012.)
Hypotheses
Ref Expression
2lplnj.l  |-  .<_  =  ( le `  K )
2lplnj.j  |-  .\/  =  ( join `  K )
2lplnj.p  |-  P  =  ( LPlanes `  K )
2lplnj.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
2lplnj  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  .\/  Y )  =  W )

Proof of Theorem 2lplnj
Dummy variables  r 
q  s  t  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2lplnj.l . . . . . . . 8  |-  .<_  =  ( le `  K )
3 2lplnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
4 eqid 2296 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 2lplnj.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 30347 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  e.  P  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r )  /\  X  =  ( ( q 
.\/  r )  .\/  s ) ) ) ) )
7 simpr 447 . . . . . . 7  |-  ( ( X  e.  ( Base `  K )  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  E. q  e.  (
Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )
86, 7syl6bi 219 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  P  ->  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) ) )
91, 2, 3, 4, 5islpln2 30347 . . . . . . 7  |-  ( K  e.  HL  ->  ( Y  e.  P  <->  ( Y  e.  ( Base `  K
)  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) ) ) ) )
10 simpr 447 . . . . . . 7  |-  ( ( Y  e.  ( Base `  K )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  E. t  e.  (
Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K )
( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )
119, 10syl6bi 219 . . . . . 6  |-  ( K  e.  HL  ->  ( Y  e.  P  ->  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) ) )
128, 11anim12d 546 . . . . 5  |-  ( K  e.  HL  ->  (
( X  e.  P  /\  Y  e.  P
)  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) ) ) )
1312imp 418 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) ) )
14133adantr3 1116 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) ) )
15143adant3 975 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) ) )
16 simpl33 1038 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  ->  X  =  ( (
q  .\/  r )  .\/  s ) )
17163ad2ant1 976 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  X  =  ( ( q  .\/  r
)  .\/  s )
)
18 simp33 993 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  Y  =  ( ( t  .\/  u
)  .\/  v )
)
1917, 18oveq12d 5892 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( X  .\/  Y )  =  ( ( ( q  .\/  r
)  .\/  s )  .\/  ( ( t  .\/  u )  .\/  v
) ) )
20 simp11 985 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  K  e.  HL )
21 simp123 1089 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  W  e.  V
)
2220, 21jca 518 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  ( K  e.  HL  /\  W  e.  V ) )
2322adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
( K  e.  HL  /\  W  e.  V ) )
24233ad2ant1 976 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( K  e.  HL  /\  W  e.  V ) )
25 simp2l 981 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  q  e.  (
Atoms `  K ) )
26 simp2rl 1024 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  r  e.  (
Atoms `  K ) )
27 simp2rr 1025 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  s  e.  (
Atoms `  K ) )
2825, 26, 273jca 1132 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )
2928adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )
30293ad2ant1 976 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )
31 simpl31 1036 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
q  =/=  r )
32313ad2ant1 976 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  q  =/=  r
)
33 simpl32 1037 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  ->  -.  s  .<_  ( q 
.\/  r ) )
34333ad2ant1 976 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  -.  s  .<_  ( q  .\/  r ) )
3532, 34jca 518 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
) ) )
36 simp1r 980 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  t  e.  (
Atoms `  K ) )
37 simp2l 981 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  u  e.  (
Atoms `  K ) )
38 simp2r 982 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  v  e.  (
Atoms `  K ) )
3936, 37, 383jca 1132 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( t  e.  ( Atoms `  K )  /\  u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) ) )
40 simp31 991 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  t  =/=  u
)
41 simp32 992 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  -.  v  .<_  ( t  .\/  u ) )
4240, 41jca 518 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u
) ) )
43 simpl13 1032 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )
44433ad2ant1 976 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )
45 breq1 4042 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  .<_  W  <->  ( (
q  .\/  r )  .\/  s )  .<_  W ) )
46 neeq1 2467 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  Y
) )
4745, 463anbi13d 1254 . . . . . . . . . . . . . . 15  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
( q  .\/  r
)  .\/  s )  .<_  W  /\  Y  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  Y
) ) )
48 breq1 4042 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  ( Y  .<_  W  <->  ( (
t  .\/  u )  .\/  v )  .<_  W ) )
49 neeq2 2468 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  (
( ( q  .\/  r )  .\/  s
)  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) )
5048, 493anbi23d 1255 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  (
( ( ( q 
.\/  r )  .\/  s )  .<_  W  /\  Y  .<_  W  /\  (
( q  .\/  r
)  .\/  s )  =/=  Y )  <->  ( (
( q  .\/  r
)  .\/  s )  .<_  W  /\  ( ( t  .\/  u ) 
.\/  v )  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) ) )
5147, 50sylan9bb 680 . . . . . . . . . . . . . 14  |-  ( ( X  =  ( ( q  .\/  r ) 
.\/  s )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) )  -> 
( ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
( q  .\/  r
)  .\/  s )  .<_  W  /\  ( ( t  .\/  u ) 
.\/  v )  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) ) )
5217, 18, 51syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( ( X 
.<_  W  /\  Y  .<_  W  /\  X  =/=  Y
)  <->  ( ( ( q  .\/  r ) 
.\/  s )  .<_  W  /\  ( ( t 
.\/  u )  .\/  v )  .<_  W  /\  ( ( q  .\/  r )  .\/  s
)  =/=  ( ( t  .\/  u ) 
.\/  v ) ) ) )
5344, 52mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( ( ( q  .\/  r ) 
.\/  s )  .<_  W  /\  ( ( t 
.\/  u )  .\/  v )  .<_  W  /\  ( ( q  .\/  r )  .\/  s
)  =/=  ( ( t  .\/  u ) 
.\/  v ) ) )
54 2lplnj.v . . . . . . . . . . . . 13  |-  V  =  ( LVols `  K )
552, 3, 4, 542lplnja 30430 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  V )  /\  (
q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
) ) )  /\  ( ( t  e.  ( Atoms `  K )  /\  u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u ) ) )  /\  ( ( ( q  .\/  r
)  .\/  s )  .<_  W  /\  ( ( t  .\/  u ) 
.\/  v )  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) )  ->  (
( ( q  .\/  r )  .\/  s
)  .\/  ( (
t  .\/  u )  .\/  v ) )  =  W )
5624, 30, 35, 39, 42, 53, 55syl321anc 1204 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( ( ( q  .\/  r ) 
.\/  s )  .\/  ( ( t  .\/  u )  .\/  v
) )  =  W )
5719, 56eqtrd 2328 . . . . . . . . . 10  |-  ( ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( X  .\/  Y )  =  W )
58573exp 1150 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
( ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  ->  ( ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) ) )
5958rexlimdvv 2686 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V )  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/= 
Y ) )  /\  ( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  X  =  ( ( q  .\/  r )  .\/  s
) ) )  /\  t  e.  ( Atoms `  K ) )  -> 
( E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) )
6059rexlimdva 2680 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
) )  /\  (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) ) )  ->  ( E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) )
61603exp 1150 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( q  e.  (
Atoms `  K )  /\  ( r  e.  (
Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  ->  ( ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r )  /\  X  =  ( ( q 
.\/  r )  .\/  s ) )  -> 
( E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) ) ) )
6261expdimp 426 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  q  e.  ( Atoms `  K )
)  ->  ( (
r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K )
)  ->  ( (
q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  -> 
( E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) ) ) )
6362rexlimdvv 2686 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  q  e.  ( Atoms `  K )
)  ->  ( E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  -> 
( E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) ) )
6463rexlimdva 2680 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  X  =  ( (
q  .\/  r )  .\/  s ) )  -> 
( E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t  .\/  u )  /\  Y  =  ( ( t 
.\/  u )  .\/  v ) )  -> 
( X  .\/  Y
)  =  W ) ) )
6564imp3a 420 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r )  /\  X  =  ( ( q 
.\/  r )  .\/  s ) )  /\  E. t  e.  ( Atoms `  K ) E. u  e.  ( Atoms `  K ) E. v  e.  ( Atoms `  K ) ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  ( X  .\/  Y )  =  W ) )
6615, 65mpd 14 1  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  Y  e.  P  /\  W  e.  V
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  .\/  Y )  =  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303   LVolsclvol 30304
This theorem is referenced by:  2lplnm2N  30432  dalem13  30487
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-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-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-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  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
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