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Theorem 2llnjN 30426
Description: The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2llnj.l  |-  .<_  =  ( le `  K )
2llnj.j  |-  .\/  =  ( join `  K )
2llnj.n  |-  N  =  ( LLines `  K )
2llnj.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
2llnjN  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  .\/  Y )  =  W )

Proof of Theorem 2llnjN
Dummy variables  r 
q  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2llnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 eqid 2438 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 2llnj.n . . . . . . . 8  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln2 30370 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  e.  N  <->  ( X  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) ) ) ) )
6 simpr 449 . . . . . . 7  |-  ( ( X  e.  ( Base `  K )  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K )
( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  ->  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) ) )
75, 6syl6bi 221 . . . . . 6  |-  ( K  e.  HL  ->  ( X  e.  N  ->  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K )
( q  =/=  r  /\  X  =  (
q  .\/  r )
) ) )
81, 2, 3, 4islln2 30370 . . . . . . 7  |-  ( K  e.  HL  ->  ( Y  e.  N  <->  ( Y  e.  ( Base `  K
)  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) ) ) ) )
9 simpr 449 . . . . . . 7  |-  ( ( Y  e.  ( Base `  K )  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) )  ->  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) ) )
108, 9syl6bi 221 . . . . . 6  |-  ( K  e.  HL  ->  ( Y  e.  N  ->  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
117, 10anim12d 548 . . . . 5  |-  ( K  e.  HL  ->  (
( X  e.  N  /\  Y  e.  N
)  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K )
( q  =/=  r  /\  X  =  (
q  .\/  r )
)  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) ) ) ) )
1211imp 420 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) )  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
13123adantr3 1119 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) )  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
14133adant3 978 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) )  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
15 simp2rr 1028 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  ->  X  =  ( q  .\/  r ) )
16 simp3rr 1032 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  ->  Y  =  ( s  .\/  t ) )
1715, 16oveq12d 6101 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( X  .\/  Y
)  =  ( ( q  .\/  r ) 
.\/  ( s  .\/  t ) ) )
18 simp13 990 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y ) )
19 breq1 4217 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  .<_  W  <->  ( q  .\/  r )  .<_  W ) )
20 neeq1 2611 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  =/=  Y  <->  ( q  .\/  r )  =/=  Y
) )
2119, 203anbi13d 1257 . . . . . . . . . . . . . 14  |-  ( X  =  ( q  .\/  r )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  ( q  .\/  r )  =/=  Y
) ) )
22 breq1 4217 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  ( Y  .<_  W  <->  ( s  .\/  t )  .<_  W ) )
23 neeq2 2612 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  (
( q  .\/  r
)  =/=  Y  <->  ( q  .\/  r )  =/=  (
s  .\/  t )
) )
2422, 233anbi23d 1258 . . . . . . . . . . . . . 14  |-  ( Y  =  ( s  .\/  t )  ->  (
( ( q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  (
q  .\/  r )  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  ( s 
.\/  t )  .<_  W  /\  ( q  .\/  r )  =/=  (
s  .\/  t )
) ) )
2521, 24sylan9bb 682 . . . . . . . . . . . . 13  |-  ( ( X  =  ( q 
.\/  r )  /\  Y  =  ( s  .\/  t ) )  -> 
( ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  ( s 
.\/  t )  .<_  W  /\  ( q  .\/  r )  =/=  (
s  .\/  t )
) ) )
2615, 16, 25syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  ( s 
.\/  t )  .<_  W  /\  ( q  .\/  r )  =/=  (
s  .\/  t )
) ) )
2718, 26mpbid 203 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( ( q  .\/  r )  .<_  W  /\  ( s  .\/  t
)  .<_  W  /\  (
q  .\/  r )  =/=  ( s  .\/  t
) ) )
28 simp11 988 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  ->  K  e.  HL )
29 simp123 1092 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  ->  W  e.  P )
30 simp2ll 1025 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
q  e.  ( Atoms `  K ) )
31 simp2lr 1026 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
r  e.  ( Atoms `  K ) )
32 simp2rl 1027 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
q  =/=  r )
33 simp3ll 1029 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
s  e.  ( Atoms `  K ) )
34 simp3lr 1030 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
t  e.  ( Atoms `  K ) )
35 simp3rl 1031 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
s  =/=  t )
36 2llnj.l . . . . . . . . . . . . . 14  |-  .<_  =  ( le `  K )
37 2llnj.p . . . . . . . . . . . . . 14  |-  P  =  ( LPlanes `  K )
3836, 2, 3, 4, 372llnjaN 30425 . . . . . . . . . . . . 13  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  (
q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  q  =/=  r
)  /\  ( s  e.  ( Atoms `  K )  /\  t  e.  ( Atoms `  K )  /\  s  =/=  t ) )  /\  ( ( q 
.\/  r )  .<_  W  /\  ( s  .\/  t )  .<_  W  /\  ( q  .\/  r
)  =/=  ( s 
.\/  t ) ) )  ->  ( (
q  .\/  r )  .\/  ( s  .\/  t
) )  =  W )
3938ex 425 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  q  =/=  r )  /\  ( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K )  /\  s  =/=  t ) )  -> 
( ( ( q 
.\/  r )  .<_  W  /\  ( s  .\/  t )  .<_  W  /\  ( q  .\/  r
)  =/=  ( s 
.\/  t ) )  ->  ( ( q 
.\/  r )  .\/  ( s  .\/  t
) )  =  W ) )
4028, 29, 30, 31, 32, 33, 34, 35, 39syl233anc 1214 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( ( ( q 
.\/  r )  .<_  W  /\  ( s  .\/  t )  .<_  W  /\  ( q  .\/  r
)  =/=  ( s 
.\/  t ) )  ->  ( ( q 
.\/  r )  .\/  ( s  .\/  t
) )  =  W ) )
4127, 40mpd 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( ( q  .\/  r )  .\/  (
s  .\/  t )
)  =  W )
4217, 41eqtrd 2470 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  (
q  .\/  r )
) )  /\  (
( s  e.  (
Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  /\  ( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )  -> 
( X  .\/  Y
)  =  W )
43423exp 1153 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  X  =  ( q  .\/  r
) ) )  -> 
( ( ( s  e.  ( Atoms `  K
)  /\  t  e.  ( Atoms `  K )
)  /\  ( s  =/=  t  /\  Y  =  ( s  .\/  t
) ) )  -> 
( X  .\/  Y
)  =  W ) ) )
44433impib 1152 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )
)  /\  ( q  =/=  r  /\  X  =  ( q  .\/  r
) ) )  -> 
( ( ( s  e.  ( Atoms `  K
)  /\  t  e.  ( Atoms `  K )
)  /\  ( s  =/=  t  /\  Y  =  ( s  .\/  t
) ) )  -> 
( X  .\/  Y
)  =  W ) )
4544exp3a 427 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )
)  /\  ( q  =/=  r  /\  X  =  ( q  .\/  r
) ) )  -> 
( ( s  e.  ( Atoms `  K )  /\  t  e.  ( Atoms `  K ) )  ->  ( ( s  =/=  t  /\  Y  =  ( s  .\/  t ) )  -> 
( X  .\/  Y
)  =  W ) ) )
4645rexlimdvv 2838 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  /\  (
q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )
)  /\  ( q  =/=  r  /\  X  =  ( q  .\/  r
) ) )  -> 
( E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) )  -> 
( X  .\/  Y
)  =  W ) )
47463exp 1153 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  -> 
( ( q  =/=  r  /\  X  =  ( q  .\/  r
) )  ->  ( E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) )  -> 
( X  .\/  Y
)  =  W ) ) ) )
4847rexlimdvv 2838 . . 3  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) )  -> 
( E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K ) ( s  =/=  t  /\  Y  =  ( s  .\/  t ) )  -> 
( X  .\/  Y
)  =  W ) ) )
4948imp3a 422 . 2  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  (
( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  X  =  ( q  .\/  r ) )  /\  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) )  ->  ( X  .\/  Y )  =  W ) )
5014, 49mpd 15 1  |-  ( ( K  e.  HL  /\  ( X  e.  N  /\  Y  e.  N  /\  W  e.  P
)  /\  ( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y
) )  ->  ( X  .\/  Y )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Atomscatm 30123   HLchlt 30210   LLinesclln 30350   LPlanesclpl 30351
This theorem is referenced by:  2llnm2N  30427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-llines 30357  df-lplanes 30358
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