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Theorem 2lplnj 30491
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 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2lplnj.l . . . . . . . 8  |-  .<_  =  ( le `  K )
3 2lplnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
4 eqid 2438 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 2lplnj.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 30407 . . . . . . 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 449 . . . . . . 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 221 . . . . . 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 30407 . . . . . . 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 449 . . . . . . 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 221 . . . . . 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 548 . . . . 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 420 . . . 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 1119 . . 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 978 . 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 1041 . . . . . . . . . . . . 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 979 . . . . . . . . . . . 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 996 . . . . . . . . . . . 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 6102 . . . . . . . . . . 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 988 . . . . . . . . . . . . . . 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 1092 . . . . . . . . . . . . . . 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 520 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 979 . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 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 979 . . . . . . . . . . . 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 1039 . . . . . . . . . . . . . 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 979 . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . . 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 979 . . . . . . . . . . . . 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 520 . . . . . . . . . . . 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 983 . . . . . . . . . . . . 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 984 . . . . . . . . . . . . 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 985 . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 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 994 . . . . . . . . . . . . 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 995 . . . . . . . . . . . . 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 520 . . . . . . . . . . . 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 1035 . . . . . . . . . . . . . 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 979 . . . . . . . . . . . . 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 4218 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  .<_  W  <->  ( (
q  .\/  r )  .\/  s )  .<_  W ) )
46 neeq1 2611 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  Y
) )
4745, 463anbi13d 1257 . . . . . . . . . . . . . . 15  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
( q  .\/  r
)  .\/  s )  .<_  W  /\  Y  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  Y
) ) )
48 breq1 4218 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  ( Y  .<_  W  <->  ( (
t  .\/  u )  .\/  v )  .<_  W ) )
49 neeq2 2612 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  (
( ( q  .\/  r )  .\/  s
)  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) )
5048, 493anbi23d 1258 . . . . . . . . . . . . . . 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 682 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 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 203 . . . . . . . . . . . 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 30490 . . . . . . . . . . . 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 1207 . . . . . . . . . . 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 2470 . . . . . . . . . 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 1153 . . . . . . . . 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 2838 . . . . . . . 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 2832 . . . . . . 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 1153 . . . . . 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 428 . . . . 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 2838 . . . 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 2832 . . 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 422 . 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 15 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363   LVolsclvol 30364
This theorem is referenced by:  2lplnm2N  30492  dalem13  30547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371
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