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Theorem 2llnjN 29756
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 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2 2llnj.j . . . . . . . 8  |-  .\/  =  ( join `  K )
3 eqid 2283 . . . . . . . 8  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 2llnj.n . . . . . . . 8  |-  N  =  ( LLines `  K )
51, 2, 3, 4islln2 29700 . . . . . . 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 29700 . . . . . . 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 5876 . . . . . . . . . 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 4026 . . . . . . . . . . . . . . 15  |-  ( X  =  ( q  .\/  r )  ->  ( X  .<_  W  <->  ( q  .\/  r )  .<_  W ) )
20 neeq1 2454 . . . . . . . . . . . . . . 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 4026 . . . . . . . . . . . . . . 15  |-  ( Y  =  ( s  .\/  t )  ->  ( Y  .<_  W  <->  ( s  .\/  t )  .<_  W ) )
23 neeq2 2455 . . . . . . . . . . . . . . 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 29755 . . . . . . . . . . . . 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 2315 . . . . . . . . 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 2673 . . . . 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 2673 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
This theorem is referenced by:  2llnm2N  29757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688
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