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Theorem 4atex2-0cOLDN 30891
Description: Same as 4atex2 30888 except that  S and 
T are zero. TODO: do we need this one or 4atex2-0aOLDN 30889 or 4atex2-0bOLDN 30890? (Contributed by NM, 27-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
4that.l  |-  .<_  =  ( le `  K )
4that.j  |-  .\/  =  ( join `  K )
4that.a  |-  A  =  ( Atoms `  K )
4that.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
4atex2-0cOLDN  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z
) ) )
Distinct variable groups:    z, r, A    H, r    .\/ , r,
z    K, r, z    .<_ , r, z    P, r, z    Q, r, z    S, r, z    W, r, z    T, r, z    z, H

Proof of Theorem 4atex2-0cOLDN
StepHypRef Expression
1 simp21l 1072 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  P  e.  A )
2 simp21r 1073 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  -.  P  .<_  W )
3 simp23 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  S  =  ( 0. `  K ) )
43oveq1d 5889 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( S  .\/  P )  =  ( ( 0. `  K
)  .\/  P )
)
5 simp32 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  T  =  ( 0. `  K ) )
65oveq1d 5889 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( T  .\/  P )  =  ( ( 0. `  K
)  .\/  P )
)
74, 6eqtr4d 2331 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  ( S  .\/  P )  =  ( T  .\/  P ) )
8 breq1 4042 . . . . 5  |-  ( z  =  P  ->  (
z  .<_  W  <->  P  .<_  W ) )
98notbid 285 . . . 4  |-  ( z  =  P  ->  ( -.  z  .<_  W  <->  -.  P  .<_  W ) )
10 oveq2 5882 . . . . 5  |-  ( z  =  P  ->  ( S  .\/  z )  =  ( S  .\/  P
) )
11 oveq2 5882 . . . . 5  |-  ( z  =  P  ->  ( T  .\/  z )  =  ( T  .\/  P
) )
1210, 11eqeq12d 2310 . . . 4  |-  ( z  =  P  ->  (
( S  .\/  z
)  =  ( T 
.\/  z )  <->  ( S  .\/  P )  =  ( T  .\/  P ) ) )
139, 12anbi12d 691 . . 3  |-  ( z  =  P  ->  (
( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z ) )  <->  ( -.  P  .<_  W  /\  ( S 
.\/  P )  =  ( T  .\/  P
) ) ) )
1413rspcev 2897 . 2  |-  ( ( P  e.  A  /\  ( -.  P  .<_  W  /\  ( S  .\/  P )  =  ( T 
.\/  P ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z
) ) )
151, 2, 7, 14syl12anc 1180 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  =  ( 0. `  K ) )  /\  ( P  =/=  Q  /\  T  =  ( 0. `  K )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( S  .\/  z )  =  ( T  .\/  z
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   0.cp0 14159   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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