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Theorem 3dim1lem5 35587
Description: Lemma for 3dim1 35588. (Contributed by NM, 26-Jul-2012.)
Hypotheses
Ref Expression
3dim0.j  |-  .\/  =  ( join `  K )
3dim0.l  |-  .<_  =  ( le `  K )
3dim0.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3dim1lem5  |-  ( ( ( u  e.  A  /\  v  e.  A  /\  w  e.  A
)  /\  ( P  =/=  u  /\  -.  v  .<_  ( P  .\/  u
)  /\  -.  w  .<_  ( ( P  .\/  u )  .\/  v
) ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P 
.\/  q )  /\  -.  s  .<_  ( ( P  .\/  q ) 
.\/  r ) ) )
Distinct variable groups:    r, q,
s, A    .\/ , r, s   
v, u, w, A, q    .\/ , q, u, v, w    u, K, v, w    .<_ , q    u, r,
v, w,  .<_ , s    P, q, r, s, u, v, w
Allowed substitution hints:    K( s, r, q)

Proof of Theorem 3dim1lem5
StepHypRef Expression
1 neeq2 2737 . . 3  |-  ( q  =  u  ->  ( P  =/=  q  <->  P  =/=  u ) )
2 oveq2 6278 . . . . 5  |-  ( q  =  u  ->  ( P  .\/  q )  =  ( P  .\/  u
) )
32breq2d 4451 . . . 4  |-  ( q  =  u  ->  (
r  .<_  ( P  .\/  q )  <->  r  .<_  ( P  .\/  u ) ) )
43notbid 292 . . 3  |-  ( q  =  u  ->  ( -.  r  .<_  ( P 
.\/  q )  <->  -.  r  .<_  ( P  .\/  u
) ) )
52oveq1d 6285 . . . . 5  |-  ( q  =  u  ->  (
( P  .\/  q
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  r ) )
65breq2d 4451 . . . 4  |-  ( q  =  u  ->  (
s  .<_  ( ( P 
.\/  q )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  r )
) )
76notbid 292 . . 3  |-  ( q  =  u  ->  ( -.  s  .<_  ( ( P  .\/  q ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  r
) ) )
81, 4, 73anbi123d 1297 . 2  |-  ( q  =  u  ->  (
( P  =/=  q  /\  -.  r  .<_  ( P 
.\/  q )  /\  -.  s  .<_  ( ( P  .\/  q ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) ) ) )
9 breq1 4442 . . . 4  |-  ( r  =  v  ->  (
r  .<_  ( P  .\/  u )  <->  v  .<_  ( P  .\/  u ) ) )
109notbid 292 . . 3  |-  ( r  =  v  ->  ( -.  r  .<_  ( P 
.\/  u )  <->  -.  v  .<_  ( P  .\/  u
) ) )
11 oveq2 6278 . . . . 5  |-  ( r  =  v  ->  (
( P  .\/  u
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  v ) )
1211breq2d 4451 . . . 4  |-  ( r  =  v  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1312notbid 292 . . 3  |-  ( r  =  v  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  v
) ) )
1410, 133anbi23d 1300 . 2  |-  ( r  =  v  ->  (
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
15 breq1 4442 . . . 4  |-  ( s  =  w  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  v )  <->  w  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1615notbid 292 . . 3  |-  ( s  =  w  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  v )  <->  -.  w  .<_  ( ( P  .\/  u )  .\/  v
) ) )
17163anbi3d 1303 . 2  |-  ( s  =  w  ->  (
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  w  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
188, 14, 17rspc3ev 3220 1  |-  ( ( ( u  e.  A  /\  v  e.  A  /\  w  e.  A
)  /\  ( P  =/=  u  /\  -.  v  .<_  ( P  .\/  u
)  /\  -.  w  .<_  ( ( P  .\/  u )  .\/  v
) ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( P  =/=  q  /\  -.  r  .<_  ( P 
.\/  q )  /\  -.  s  .<_  ( ( P  .\/  q ) 
.\/  r ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   lecple 14791   joincjn 15772   Atomscatm 35385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  3dim1  35588
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