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Theorem hlsuprexch 30192
Description: A Hilbert lattice has the superposition and exchange properties. (Contributed by NM, 13-Nov-2011.)
Hypotheses
Ref Expression
hlsuprexch.b  |-  B  =  ( Base `  K
)
hlsuprexch.l  |-  .<_  =  ( le `  K )
hlsuprexch.j  |-  .\/  =  ( join `  K )
hlsuprexch.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlsuprexch  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  =/= 
Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/= 
Q  /\  z  .<_  ( P  .\/  Q ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) )  ->  Q  .<_  ( z 
.\/  P ) ) ) )
Distinct variable groups:    z, A    z, B    z, K    z, P    z, Q
Allowed substitution hints:    .\/ ( z)    .<_ ( z)

Proof of Theorem hlsuprexch
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hlsuprexch.b . . . . 5  |-  B  =  ( Base `  K
)
2 hlsuprexch.l . . . . 5  |-  .<_  =  ( le `  K )
3 eqid 2296 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
4 hlsuprexch.j . . . . 5  |-  .\/  =  ( join `  K )
5 eqid 2296 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2296 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 hlsuprexch.a . . . . 5  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 30165 . . . 4  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat
)  /\  ( A. x  e.  A  A. y  e.  A  (
( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  x ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( ( ( 0. `  K ) ( lt `  K
) x  /\  x
( lt `  K
) y )  /\  ( y ( lt
`  K ) z  /\  z ( lt
`  K ) ( 1. `  K ) ) ) ) ) )
9 simprl 732 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  AtLat )  /\  ( A. x  e.  A  A. y  e.  A  ( ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) )  /\  A. z  e.  B  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  ->  y  .<_  ( z 
.\/  x ) ) )  /\  E. x  e.  B  E. y  e.  B  E. z  e.  B  ( (
( 0. `  K
) ( lt `  K ) x  /\  x ( lt `  K ) y )  /\  ( y ( lt `  K ) z  /\  z ( lt `  K ) ( 1. `  K
) ) ) ) )  ->  A. x  e.  A  A. y  e.  A  ( (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  x ) ) ) )
108, 9sylbi 187 . . 3  |-  ( K  e.  HL  ->  A. x  e.  A  A. y  e.  A  ( (
x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  /\  A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  x ) ) ) )
11 neeq1 2467 . . . . . 6  |-  ( x  =  P  ->  (
x  =/=  y  <->  P  =/=  y ) )
12 neeq2 2468 . . . . . . . 8  |-  ( x  =  P  ->  (
z  =/=  x  <->  z  =/=  P ) )
13 oveq1 5881 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .\/  y )  =  ( P  .\/  y ) )
1413breq2d 4051 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .<_  ( x  .\/  y )  <->  z  .<_  ( P  .\/  y ) ) )
1512, 143anbi13d 1254 . . . . . . 7  |-  ( x  =  P  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) ) )
1615rexbidv 2577 . . . . . 6  |-  ( x  =  P  ->  ( E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) )  <->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) ) )
1711, 16imbi12d 311 . . . . 5  |-  ( x  =  P  ->  (
( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) ) )  <->  ( P  =/=  y  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P  .\/  y ) ) ) ) )
18 breq1 4042 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .<_  z  <->  P  .<_  z ) )
1918notbid 285 . . . . . . . 8  |-  ( x  =  P  ->  ( -.  x  .<_  z  <->  -.  P  .<_  z ) )
20 breq1 4042 . . . . . . . 8  |-  ( x  =  P  ->  (
x  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  y ) ) )
2119, 20anbi12d 691 . . . . . . 7  |-  ( x  =  P  ->  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) ) ) )
22 oveq2 5882 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .\/  x )  =  ( z  .\/  P ) )
2322breq2d 4051 . . . . . . 7  |-  ( x  =  P  ->  (
y  .<_  ( z  .\/  x )  <->  y  .<_  ( z  .\/  P ) ) )
2421, 23imbi12d 311 . . . . . 6  |-  ( x  =  P  ->  (
( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  x ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  P ) ) ) )
2524ralbidv 2576 . . . . 5  |-  ( x  =  P  ->  ( A. z  e.  B  ( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  x ) )  <->  A. z  e.  B  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  P ) ) ) )
2617, 25anbi12d 691 . . . 4  |-  ( x  =  P  ->  (
( ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) )  /\  A. z  e.  B  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  ->  y  .<_  ( z 
.\/  x ) ) )  <->  ( ( P  =/=  y  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P  .\/  y ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) )  ->  y  .<_  ( z 
.\/  P ) ) ) ) )
27 neeq2 2468 . . . . . 6  |-  ( y  =  Q  ->  ( P  =/=  y  <->  P  =/=  Q ) )
28 neeq2 2468 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  =/=  y  <->  z  =/=  Q ) )
29 oveq2 5882 . . . . . . . . 9  |-  ( y  =  Q  ->  ( P  .\/  y )  =  ( P  .\/  Q
) )
3029breq2d 4051 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  .<_  ( P  .\/  y )  <->  z  .<_  ( P  .\/  Q ) ) )
3128, 303anbi23d 1255 . . . . . . 7  |-  ( y  =  Q  ->  (
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) ) )
3231rexbidv 2577 . . . . . 6  |-  ( y  =  Q  ->  ( E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) )  <->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) ) )
3327, 32imbi12d 311 . . . . 5  |-  ( y  =  Q  ->  (
( P  =/=  y  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) )  <->  ( P  =/= 
Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/= 
Q  /\  z  .<_  ( P  .\/  Q ) ) ) ) )
34 oveq2 5882 . . . . . . . . 9  |-  ( y  =  Q  ->  (
z  .\/  y )  =  ( z  .\/  Q ) )
3534breq2d 4051 . . . . . . . 8  |-  ( y  =  Q  ->  ( P  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  Q ) ) )
3635anbi2d 684 . . . . . . 7  |-  ( y  =  Q  ->  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) ) ) )
37 breq1 4042 . . . . . . 7  |-  ( y  =  Q  ->  (
y  .<_  ( z  .\/  P )  <->  Q  .<_  ( z 
.\/  P ) ) )
3836, 37imbi12d 311 . . . . . 6  |-  ( y  =  Q  ->  (
( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  P ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  Q ) )  ->  Q  .<_  ( z  .\/  P
) ) ) )
3938ralbidv 2576 . . . . 5  |-  ( y  =  Q  ->  ( A. z  e.  B  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  P ) )  <->  A. z  e.  B  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  Q ) )  ->  Q  .<_  ( z  .\/  P ) ) ) )
4033, 39anbi12d 691 . . . 4  |-  ( y  =  Q  ->  (
( ( P  =/=  y  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P  .\/  y ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) )  ->  y  .<_  ( z 
.\/  P ) ) )  <->  ( ( P  =/=  Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/= 
Q  /\  z  .<_  ( P  .\/  Q ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) )  ->  Q  .<_  ( z 
.\/  P ) ) ) ) )
4126, 40rspc2v 2903 . . 3  |-  ( ( P  e.  A  /\  Q  e.  A )  ->  ( A. x  e.  A  A. y  e.  A  ( ( x  =/=  y  ->  E. z  e.  A  ( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x  .\/  y ) ) )  /\  A. z  e.  B  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  ->  y  .<_  ( z 
.\/  x ) ) )  ->  ( ( P  =/=  Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/= 
Q  /\  z  .<_  ( P  .\/  Q ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) )  ->  Q  .<_  ( z 
.\/  P ) ) ) ) )
4210, 41mpan9 455 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  (
( P  =/=  Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) )  /\  A. z  e.  B  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  Q ) )  ->  Q  .<_  ( z  .\/  P
) ) ) )
43423impb 1147 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  =/= 
Q  ->  E. z  e.  A  ( z  =/=  P  /\  z  =/= 
Q  /\  z  .<_  ( P  .\/  Q ) ) )  /\  A. z  e.  B  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) )  ->  Q  .<_  ( z 
.\/  P ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   ltcplt 14091   joincjn 14094   0.cp0 14159   1.cp1 14160   CLatccla 14229   OMLcoml 29987   Atomscatm 30075   AtLatcal 30076   HLchlt 30162
This theorem is referenced by:  hlsupr  30197
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-ne 2461  df-ral 2561  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  df-cvlat 30134  df-hlat 30163
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