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Theorem 2llnjN 30378
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 2296 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2llnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 eqid 2296 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 2llnj.n . . . . . . . 8  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln2 30322 . . . . . . 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 447 . . . . . . 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 219 . . . . . 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 30322 . . . . . . 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 447 . . . . . . 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 219 . . . . . 6  |-  ( K  e.  HL  ->  ( Y  e.  N  ->  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
117, 10anim12d 546 . . . . 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 418 . . . 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 1116 . . 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 975 . 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 1025 . . . . . . . . . . 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 1029 . . . . . . . . . . 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 5892 . . . . . . . . . 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 987 . . . . . . . . . . . 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 4042 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  .<_  W  <->  ( q  .\/  r )  .<_  W ) )
20 neeq1 2467 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  =/=  Y  <->  ( q  .\/  r )  =/=  Y
) )
2119, 203anbi13d 1254 . . . . . . . . . . . . . 14  |-  ( X  =  ( q  .\/  r )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  ( q  .\/  r )  =/=  Y
) ) )
22 breq1 4042 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  ( Y  .<_  W  <->  ( s  .\/  t )  .<_  W ) )
23 neeq2 2468 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  (
( q  .\/  r
)  =/=  Y  <->  ( q  .\/  r )  =/=  (
s  .\/  t )
) )
2422, 233anbi23d 1255 . . . . . . . . . . . . . 14  |-  ( Y  =  ( s  .\/  t )  ->  (
( ( q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  (
q  .\/  r )  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  ( s 
.\/  t )  .<_  W  /\  ( q  .\/  r )  =/=  (
s  .\/  t )
) ) )
2521, 24sylan9bb 680 . . . . . . . . . . . . 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 642 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 985 . . . . . . . . . . . 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 1089 . . . . . . . . . . . 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 1022 . . . . . . . . . . . 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 1023 . . . . . . . . . . . 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 1024 . . . . . . . . . . . 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 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 )
) ) )  -> 
s  e.  ( Atoms `  K ) )
34 simp3lr 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 )
) ) )  -> 
t  e.  ( Atoms `  K ) )
35 simp3rl 1028 . . . . . . . . . . . 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 30377 . . . . . . . . . . . . 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 423 . . . . . . . . . . . 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 1211 . . . . . . . . . . 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 14 . . . . . . . . . 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 2328 . . . . . . . . 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 1150 . . . . . . . 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 1149 . . . . . . 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 425 . . . . . 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 2686 . . . . 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 1150 . . . 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 2686 . . 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 420 . 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 14 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 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   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  2llnm2N  30379
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-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
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