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Theorem hlsuprexch 33121
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 2443 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
4 hlsuprexch.j . . . . 5  |-  .\/  =  ( join `  K )
5 eqid 2443 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2443 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 hlsuprexch.a . . . . 5  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 33094 . . . 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 755 . . . 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 195 . . 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 2644 . . . . . 6  |-  ( x  =  P  ->  (
x  =/=  y  <->  P  =/=  y ) )
12 neeq2 2646 . . . . . . . 8  |-  ( x  =  P  ->  (
z  =/=  x  <->  z  =/=  P ) )
13 oveq1 6119 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .\/  y )  =  ( P  .\/  y ) )
1413breq2d 4325 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .<_  ( x  .\/  y )  <->  z  .<_  ( P  .\/  y ) ) )
1512, 143anbi13d 1291 . . . . . . 7  |-  ( x  =  P  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) ) )
1615rexbidv 2757 . . . . . 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 320 . . . . 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 4316 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .<_  z  <->  P  .<_  z ) )
1918notbid 294 . . . . . . . 8  |-  ( x  =  P  ->  ( -.  x  .<_  z  <->  -.  P  .<_  z ) )
20 breq1 4316 . . . . . . . 8  |-  ( x  =  P  ->  (
x  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  y ) ) )
2119, 20anbi12d 710 . . . . . . 7  |-  ( x  =  P  ->  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) ) ) )
22 oveq2 6120 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .\/  x )  =  ( z  .\/  P ) )
2322breq2d 4325 . . . . . . 7  |-  ( x  =  P  ->  (
y  .<_  ( z  .\/  x )  <->  y  .<_  ( z  .\/  P ) ) )
2421, 23imbi12d 320 . . . . . 6  |-  ( x  =  P  ->  (
( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  x ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  P ) ) ) )
2524ralbidv 2756 . . . . 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 710 . . . 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 2646 . . . . . 6  |-  ( y  =  Q  ->  ( P  =/=  y  <->  P  =/=  Q ) )
28 neeq2 2646 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  =/=  y  <->  z  =/=  Q ) )
29 oveq2 6120 . . . . . . . . 9  |-  ( y  =  Q  ->  ( P  .\/  y )  =  ( P  .\/  Q
) )
3029breq2d 4325 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  .<_  ( P  .\/  y )  <->  z  .<_  ( P  .\/  Q ) ) )
3128, 303anbi23d 1292 . . . . . . 7  |-  ( y  =  Q  ->  (
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) ) )
3231rexbidv 2757 . . . . . 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 320 . . . . 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 6120 . . . . . . . . 9  |-  ( y  =  Q  ->  (
z  .\/  y )  =  ( z  .\/  Q ) )
3534breq2d 4325 . . . . . . . 8  |-  ( y  =  Q  ->  ( P  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  Q ) ) )
3635anbi2d 703 . . . . . . 7  |-  ( y  =  Q  ->  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) ) ) )
37 breq1 4316 . . . . . . 7  |-  ( y  =  Q  ->  (
y  .<_  ( z  .\/  P )  <->  Q  .<_  ( z 
.\/  P ) ) )
3836, 37imbi12d 320 . . . . . 6  |-  ( y  =  Q  ->  (
( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  P ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  Q ) )  ->  Q  .<_  ( z  .\/  P
) ) ) )
3938ralbidv 2756 . . . . 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 710 . . . 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 3100 . . 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 469 . 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 1183 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   class class class wbr 4313   ` cfv 5439  (class class class)co 6112   Basecbs 14195   lecple 14266   ltcplt 15132   joincjn 15135   0.cp0 15228   1.cp1 15229   CLatccla 15298   OMLcoml 32916   Atomscatm 33004   AtLatcal 33005   HLchlt 33091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-iota 5402  df-fv 5447  df-ov 6115  df-cvlat 33063  df-hlat 33092
This theorem is referenced by:  hlsupr  33126
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