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Theorem 2llnjN 30049
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 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2llnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 eqid 2404 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 2llnj.n . . . . . . . 8  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln2 29993 . . . . . . 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 448 . . . . . . 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 220 . . . . . 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 29993 . . . . . . 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 448 . . . . . . 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 220 . . . . . 6  |-  ( K  e.  HL  ->  ( Y  e.  N  ->  E. s  e.  ( Atoms `  K ) E. t  e.  ( Atoms `  K )
( s  =/=  t  /\  Y  =  (
s  .\/  t )
) ) )
117, 10anim12d 547 . . . . 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 419 . . . 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 1118 . . 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 977 . 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 1027 . . . . . . . . . . 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 1031 . . . . . . . . . . 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 6058 . . . . . . . . . 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 989 . . . . . . . . . . . 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 4175 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  .<_  W  <->  ( q  .\/  r )  .<_  W ) )
20 neeq1 2575 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  =/=  Y  <->  ( q  .\/  r )  =/=  Y
) )
2119, 203anbi13d 1256 . . . . . . . . . . . . . 14  |-  ( X  =  ( q  .\/  r )  ->  (
( X  .<_  W  /\  Y  .<_  W  /\  X  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  ( q  .\/  r )  =/=  Y
) ) )
22 breq1 4175 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  ( Y  .<_  W  <->  ( s  .\/  t )  .<_  W ) )
23 neeq2 2576 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  (
( q  .\/  r
)  =/=  Y  <->  ( q  .\/  r )  =/=  (
s  .\/  t )
) )
2422, 233anbi23d 1257 . . . . . . . . . . . . . 14  |-  ( Y  =  ( s  .\/  t )  ->  (
( ( q  .\/  r )  .<_  W  /\  Y  .<_  W  /\  (
q  .\/  r )  =/=  Y )  <->  ( (
q  .\/  r )  .<_  W  /\  ( s 
.\/  t )  .<_  W  /\  ( q  .\/  r )  =/=  (
s  .\/  t )
) ) )
2521, 24sylan9bb 681 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 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 )
) ) )  ->  K  e.  HL )
29 simp123 1091 . . . . . . . . . . . 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 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  e.  ( Atoms `  K ) )
31 simp2lr 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 )
) ) )  -> 
r  e.  ( Atoms `  K ) )
32 simp2rl 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 )
) ) )  -> 
q  =/=  r )
33 simp3ll 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  e.  ( Atoms `  K ) )
34 simp3lr 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 )
) ) )  -> 
t  e.  ( Atoms `  K ) )
35 simp3rl 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 )
) ) )  -> 
s  =/=  t )
36 2llnj.l . . . . . . . . . . . . . 14  |-  .<_  =  ( le `  K )
37 2llnj.p . . . . . . . . . . . . . 14  |-  P  =  ( LPlanes `  K )
3836, 2, 3, 4, 372llnjaN 30048 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 1213 . . . . . . . . . . 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 2436 . . . . . . . . 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 1152 . . . . . . . 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 1151 . . . . . . 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 426 . . . . . 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 2796 . . . . 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 1152 . . . 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 2796 . . 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 421 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Atomscatm 29746   HLchlt 29833   LLinesclln 29973   LPlanesclpl 29974
This theorem is referenced by:  2llnm2N  30050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981
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