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Theorem hlsuprexch 29570
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 2283 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
4 hlsuprexch.j . . . . 5  |-  .\/  =  ( join `  K )
5 eqid 2283 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2283 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 hlsuprexch.a . . . . 5  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 29543 . . . 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 2454 . . . . . 6  |-  ( x  =  P  ->  (
x  =/=  y  <->  P  =/=  y ) )
12 neeq2 2455 . . . . . . . 8  |-  ( x  =  P  ->  (
z  =/=  x  <->  z  =/=  P ) )
13 oveq1 5865 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .\/  y )  =  ( P  .\/  y ) )
1413breq2d 4035 . . . . . . . 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 2564 . . . . . 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 4026 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .<_  z  <->  P  .<_  z ) )
1918notbid 285 . . . . . . . 8  |-  ( x  =  P  ->  ( -.  x  .<_  z  <->  -.  P  .<_  z ) )
20 breq1 4026 . . . . . . . 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 5866 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .\/  x )  =  ( z  .\/  P ) )
2322breq2d 4035 . . . . . . 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 2563 . . . . 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 2455 . . . . . 6  |-  ( y  =  Q  ->  ( P  =/=  y  <->  P  =/=  Q ) )
28 neeq2 2455 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  =/=  y  <->  z  =/=  Q ) )
29 oveq2 5866 . . . . . . . . 9  |-  ( y  =  Q  ->  ( P  .\/  y )  =  ( P  .\/  Q
) )
3029breq2d 4035 . . . . . . . 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 2564 . . . . . 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 5866 . . . . . . . . 9  |-  ( y  =  Q  ->  (
z  .\/  y )  =  ( z  .\/  Q ) )
3534breq2d 4035 . . . . . . . 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 4026 . . . . . . 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 2563 . . . . 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 2890 . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   0.cp0 14143   1.cp1 14144   CLatccla 14213   OMLcoml 29365   Atomscatm 29453   AtLatcal 29454   HLchlt 29540
This theorem is referenced by:  hlsupr  29575
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-ral 2548  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  df-cvlat 29512  df-hlat 29541
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