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Theorem 3dim1lem5 29655
Description: Lemma for 3dim1 29656. (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 2455 . . 3  |-  ( q  =  u  ->  ( P  =/=  q  <->  P  =/=  u ) )
2 oveq2 5866 . . . . 5  |-  ( q  =  u  ->  ( P  .\/  q )  =  ( P  .\/  u
) )
32breq2d 4035 . . . 4  |-  ( q  =  u  ->  (
r  .<_  ( P  .\/  q )  <->  r  .<_  ( P  .\/  u ) ) )
43notbid 285 . . 3  |-  ( q  =  u  ->  ( -.  r  .<_  ( P 
.\/  q )  <->  -.  r  .<_  ( P  .\/  u
) ) )
52oveq1d 5873 . . . . 5  |-  ( q  =  u  ->  (
( P  .\/  q
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  r ) )
65breq2d 4035 . . . 4  |-  ( q  =  u  ->  (
s  .<_  ( ( P 
.\/  q )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  r )
) )
76notbid 285 . . 3  |-  ( q  =  u  ->  ( -.  s  .<_  ( ( P  .\/  q ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  r
) ) )
81, 4, 73anbi123d 1252 . 2  |-  ( q  =  u  ->  (
( P  =/=  q  /\  -.  r  .<_  ( P 
.\/  q )  /\  -.  s  .<_  ( ( P  .\/  q ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) ) ) )
9 breq1 4026 . . . 4  |-  ( r  =  v  ->  (
r  .<_  ( P  .\/  u )  <->  v  .<_  ( P  .\/  u ) ) )
109notbid 285 . . 3  |-  ( r  =  v  ->  ( -.  r  .<_  ( P 
.\/  u )  <->  -.  v  .<_  ( P  .\/  u
) ) )
11 oveq2 5866 . . . . 5  |-  ( r  =  v  ->  (
( P  .\/  u
)  .\/  r )  =  ( ( P 
.\/  u )  .\/  v ) )
1211breq2d 4035 . . . 4  |-  ( r  =  v  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  r )  <->  s  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1312notbid 285 . . 3  |-  ( r  =  v  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  r )  <->  -.  s  .<_  ( ( P  .\/  u )  .\/  v
) ) )
1410, 133anbi23d 1255 . 2  |-  ( r  =  v  ->  (
( P  =/=  u  /\  -.  r  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  r ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
15 breq1 4026 . . . 4  |-  ( s  =  w  ->  (
s  .<_  ( ( P 
.\/  u )  .\/  v )  <->  w  .<_  ( ( P  .\/  u
)  .\/  v )
) )
1615notbid 285 . . 3  |-  ( s  =  w  ->  ( -.  s  .<_  ( ( P  .\/  u ) 
.\/  v )  <->  -.  w  .<_  ( ( P  .\/  u )  .\/  v
) ) )
17163anbi3d 1258 . 2  |-  ( s  =  w  ->  (
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  s  .<_  ( ( P  .\/  u ) 
.\/  v ) )  <-> 
( P  =/=  u  /\  -.  v  .<_  ( P 
.\/  u )  /\  -.  w  .<_  ( ( P  .\/  u ) 
.\/  v ) ) ) )
188, 14, 17rspc3ev 2894 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 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   lecple 13215   joincjn 14078   Atomscatm 29453
This theorem is referenced by:  3dim1  29656
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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