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Theorem 3dim1lem5 30325
Description: Lemma for 3dim1 30326. (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 2612 . . 3  |-  ( q  =  u  ->  ( P  =/=  q  <->  P  =/=  u ) )
2 oveq2 6091 . . . . 5  |-  ( q  =  u  ->  ( P  .\/  q )  =  ( P  .\/  u
) )
32breq2d 4226 . . . 4  |-  ( q  =  u  ->  (
r  .<_  ( P  .\/  q )  <->  r  .<_  ( P  .\/  u ) ) )
43notbid 287 . . 3  |-  ( q  =  u  ->  ( -.  r  .<_  ( P 
.\/  q )  <->  -.  r  .<_  ( P  .\/  u
) ) )
52oveq1d 6098 . . . . 5  |-  ( q  =  u  ->  (
( P  .\/  q
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  r ) )
65breq2d 4226 . . . 4  |-  ( q  =  u  ->  (
s  .<_  ( ( P 
.\/  q )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  r )
) )
76notbid 287 . . 3  |-  ( q  =  u  ->  ( -.  s  .<_  ( ( P  .\/  q ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  r
) ) )
81, 4, 73anbi123d 1255 . 2  |-  ( q  =  u  ->  (
( P  =/=  q  /\  -.  r  .<_  ( P 
.\/  q )  /\  -.  s  .<_  ( ( P  .\/  q ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) ) ) )
9 breq1 4217 . . . 4  |-  ( r  =  v  ->  (
r  .<_  ( P  .\/  u )  <->  v  .<_  ( P  .\/  u ) ) )
109notbid 287 . . 3  |-  ( r  =  v  ->  ( -.  r  .<_  ( P 
.\/  u )  <->  -.  v  .<_  ( P  .\/  u
) ) )
11 oveq2 6091 . . . . 5  |-  ( r  =  v  ->  (
( P  .\/  u
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  v ) )
1211breq2d 4226 . . . 4  |-  ( r  =  v  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1312notbid 287 . . 3  |-  ( r  =  v  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  v
) ) )
1410, 133anbi23d 1258 . 2  |-  ( r  =  v  ->  (
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
15 breq1 4217 . . . 4  |-  ( s  =  w  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  v )  <->  w  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1615notbid 287 . . 3  |-  ( s  =  w  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  v )  <->  -.  w  .<_  ( ( P  .\/  u )  .\/  v
) ) )
17163anbi3d 1261 . 2  |-  ( s  =  w  ->  (
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  w  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
188, 14, 17rspc3ev 3064 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ 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   lecple 13538   joincjn 14403   Atomscatm 30123
This theorem is referenced by:  3dim1  30326
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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086
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