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Theorem 2lplnj 33097
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 2423 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2lplnj.l . . . . . . . 8  |-  .<_  =  ( le `  K )
3 2lplnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
4 eqid 2423 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 2lplnj.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
61, 2, 3, 4, 5islpln2 33013 . . . . . . 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 462 . . . . . . 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 231 . . . . . 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 33013 . . . . . . 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 462 . . . . . . 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 231 . . . . . 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 565 . . . . 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 430 . . . 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 1166 . . 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 1025 . 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 1088 . . . . . . . . . . . . 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 1026 . . . . . . . . . . . 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 1043 . . . . . . . . . . . 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 6262 . . . . . . . . . . 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 1035 . . . . . . . . . . . . . . 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 1139 . . . . . . . . . . . . . . 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 534 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 1026 . . . . . . . . . . . 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 1031 . . . . . . . . . . . . . . 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 1074 . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 1026 . . . . . . . . . . . 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 1086 . . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . 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 1087 . . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . 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 534 . . . . . . . . . . . 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 1030 . . . . . . . . . . . . 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 1031 . . . . . . . . . . . . 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 1032 . . . . . . . . . . . . 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 1185 . . . . . . . . . . . 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 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 ) )  /\  ( u  e.  ( Atoms `  K )  /\  v  e.  ( Atoms `  K ) )  /\  ( t  =/=  u  /\  -.  v  .<_  ( t 
.\/  u )  /\  Y  =  ( (
t  .\/  u )  .\/  v ) ) )  ->  t  =/=  u
)
41 simp32 1042 . . . . . . . . . . . . 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 534 . . . . . . . . . . . 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 1082 . . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . 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 4364 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  .<_  W  <->  ( (
q  .\/  r )  .\/  s )  .<_  W ) )
46 neeq1 2658 . . . . . . . . . . . . . . . 16  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  ( X  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  Y
) )
4745, 463anbi13d 1337 . . . . . . . . . . . . . . 15  |-  ( X  =  ( ( q 
.\/  r )  .\/  s )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
( q  .\/  r
)  .\/  s )  .<_  W  /\  Y  .<_  W  /\  ( ( q 
.\/  r )  .\/  s )  =/=  Y
) ) )
48 breq1 4364 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  ( Y  .<_  W  <->  ( (
t  .\/  u )  .\/  v )  .<_  W ) )
49 neeq2 2659 . . . . . . . . . . . . . . . 16  |-  ( Y  =  ( ( t 
.\/  u )  .\/  v )  ->  (
( ( q  .\/  r )  .\/  s
)  =/=  Y  <->  ( (
q  .\/  r )  .\/  s )  =/=  (
( t  .\/  u
)  .\/  v )
) )
5048, 493anbi23d 1338 . . . . . . . . . . . . . . 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 704 . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . 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 213 . . . . . . . . . . . 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 33096 . . . . . . . . . . . 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 1286 . . . . . . . . . . 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 2457 . . . . . . . . . 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 1204 . . . . . . . . 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 2857 . . . . . . . 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 2851 . . . . . . 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 1204 . . . . . 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 438 . . . . 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 2857 . . . 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 2851 . . 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 ) ) )
6564impd 432 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2594   E.wrex 2710   class class class wbr 4361   ` cfv 5539  (class class class)co 6244   Basecbs 15059   lecple 15135   joincjn 16127   Atomscatm 32741   HLchlt 32828   LPlanesclpl 32969   LVolsclvol 32970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-ral 2714  df-rex 2715  df-reu 2716  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4158  df-iun 4239  df-br 4362  df-opab 4421  df-mpt 4422  df-id 4706  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-riota 6206  df-ov 6247  df-oprab 6248  df-preset 16111  df-poset 16129  df-plt 16142  df-lub 16158  df-glb 16159  df-join 16160  df-meet 16161  df-p0 16223  df-lat 16230  df-clat 16292  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977
This theorem is referenced by:  2lplnm2N  33098  dalem13  33153
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