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Theorem hlsuprexch 30178
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 2436 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
4 hlsuprexch.j . . . . 5  |-  .\/  =  ( join `  K )
5 eqid 2436 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2436 . . . . 5  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 hlsuprexch.a . . . . 5  |-  A  =  ( Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat2 30151 . . . 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 733 . . . 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 188 . . 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 2609 . . . . . 6  |-  ( x  =  P  ->  (
x  =/=  y  <->  P  =/=  y ) )
12 neeq2 2610 . . . . . . . 8  |-  ( x  =  P  ->  (
z  =/=  x  <->  z  =/=  P ) )
13 oveq1 6088 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .\/  y )  =  ( P  .\/  y ) )
1413breq2d 4224 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .<_  ( x  .\/  y )  <->  z  .<_  ( P  .\/  y ) ) )
1512, 143anbi13d 1256 . . . . . . 7  |-  ( x  =  P  ->  (
( z  =/=  x  /\  z  =/=  y  /\  z  .<_  ( x 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) ) ) )
1615rexbidv 2726 . . . . . 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 312 . . . . 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 4215 . . . . . . . . 9  |-  ( x  =  P  ->  (
x  .<_  z  <->  P  .<_  z ) )
1918notbid 286 . . . . . . . 8  |-  ( x  =  P  ->  ( -.  x  .<_  z  <->  -.  P  .<_  z ) )
20 breq1 4215 . . . . . . . 8  |-  ( x  =  P  ->  (
x  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  y ) ) )
2119, 20anbi12d 692 . . . . . . 7  |-  ( x  =  P  ->  (
( -.  x  .<_  z  /\  x  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) ) ) )
22 oveq2 6089 . . . . . . . 8  |-  ( x  =  P  ->  (
z  .\/  x )  =  ( z  .\/  P ) )
2322breq2d 4224 . . . . . . 7  |-  ( x  =  P  ->  (
y  .<_  ( z  .\/  x )  <->  y  .<_  ( z  .\/  P ) ) )
2421, 23imbi12d 312 . . . . . 6  |-  ( x  =  P  ->  (
( ( -.  x  .<_  z  /\  x  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  x ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  -> 
y  .<_  ( z  .\/  P ) ) ) )
2524ralbidv 2725 . . . . 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 692 . . . 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 2610 . . . . . 6  |-  ( y  =  Q  ->  ( P  =/=  y  <->  P  =/=  Q ) )
28 neeq2 2610 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  =/=  y  <->  z  =/=  Q ) )
29 oveq2 6089 . . . . . . . . 9  |-  ( y  =  Q  ->  ( P  .\/  y )  =  ( P  .\/  Q
) )
3029breq2d 4224 . . . . . . . 8  |-  ( y  =  Q  ->  (
z  .<_  ( P  .\/  y )  <->  z  .<_  ( P  .\/  Q ) ) )
3128, 303anbi23d 1257 . . . . . . 7  |-  ( y  =  Q  ->  (
( z  =/=  P  /\  z  =/=  y  /\  z  .<_  ( P 
.\/  y ) )  <-> 
( z  =/=  P  /\  z  =/=  Q  /\  z  .<_  ( P 
.\/  Q ) ) ) )
3231rexbidv 2726 . . . . . 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 312 . . . . 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 6089 . . . . . . . . 9  |-  ( y  =  Q  ->  (
z  .\/  y )  =  ( z  .\/  Q ) )
3534breq2d 4224 . . . . . . . 8  |-  ( y  =  Q  ->  ( P  .<_  ( z  .\/  y )  <->  P  .<_  ( z  .\/  Q ) ) )
3635anbi2d 685 . . . . . . 7  |-  ( y  =  Q  ->  (
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  y ) )  <-> 
( -.  P  .<_  z  /\  P  .<_  ( z 
.\/  Q ) ) ) )
37 breq1 4215 . . . . . . 7  |-  ( y  =  Q  ->  (
y  .<_  ( z  .\/  P )  <->  Q  .<_  ( z 
.\/  P ) ) )
3836, 37imbi12d 312 . . . . . 6  |-  ( y  =  Q  ->  (
( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  y ) )  ->  y  .<_  ( z  .\/  P ) )  <->  ( ( -.  P  .<_  z  /\  P  .<_  ( z  .\/  Q ) )  ->  Q  .<_  ( z  .\/  P
) ) ) )
3938ralbidv 2725 . . . . 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 692 . . . 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 3058 . . 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 456 . 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 1149 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   joincjn 14401   0.cp0 14466   1.cp1 14467   CLatccla 14536   OMLcoml 29973   Atomscatm 30061   AtLatcal 30062   HLchlt 30148
This theorem is referenced by:  hlsupr  30183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-cvlat 30120  df-hlat 30149
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