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Theorem hlsuprexch 34578
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 2467 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
4 hlsuprexch.j . . . . 5  |-  .\/  =  ( join `  K )
5 eqid 2467 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2467 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 hlsuprexch.a . . . . 5  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 34551 . . . 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 2748 . . . . . 6  |-  ( x  =  P  ->  (
x  =/=  y  <->  P  =/=  y ) )
12 neeq2 2750 . . . . . . . 8  |-  ( x  =  P  ->  (
z  =/=  x  <->  z  =/=  P ) )
13 oveq1 6302 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .\/  y )  =  ( P  .\/  y ) )
1413breq2d 4465 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .<_  ( x  .\/  y )  <->  z  .<_  ( P  .\/  y ) ) )
1512, 143anbi13d 1301 . . . . . . 7  |-  ( x  =  P  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) ) )
1615rexbidv 2978 . . . . . 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 4456 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .<_  z  <->  P  .<_  z ) )
1918notbid 294 . . . . . . . 8  |-  ( x  =  P  ->  ( -.  x  .<_  z  <->  -.  P  .<_  z ) )
20 breq1 4456 . . . . . . . 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 6303 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .\/  x )  =  ( z  .\/  P ) )
2322breq2d 4465 . . . . . . 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 2906 . . . . 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 2750 . . . . . 6  |-  ( y  =  Q  ->  ( P  =/=  y  <->  P  =/=  Q ) )
28 neeq2 2750 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  =/=  y  <->  z  =/=  Q ) )
29 oveq2 6303 . . . . . . . . 9  |-  ( y  =  Q  ->  ( P  .\/  y )  =  ( P  .\/  Q
) )
3029breq2d 4465 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  .<_  ( P  .\/  y )  <->  z  .<_  ( P  .\/  Q ) ) )
3128, 303anbi23d 1302 . . . . . . 7  |-  ( y  =  Q  ->  (
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) ) )
3231rexbidv 2978 . . . . . 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 6303 . . . . . . . . 9  |-  ( y  =  Q  ->  (
z  .\/  y )  =  ( z  .\/  Q ) )
3534breq2d 4465 . . . . . . . 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 4456 . . . . . . 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 2906 . . . . 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 3228 . . 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 1192 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   ltcplt 15445   joincjn 15448   0.cp0 15541   1.cp1 15542   CLatccla 15611   OMLcoml 34373   Atomscatm 34461   AtLatcal 34462   HLchlt 34548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-cvlat 34520  df-hlat 34549
This theorem is referenced by:  hlsupr  34583
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