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Theorem hstel2 23714
Description: Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hstel2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( ( ( S `  A ) 
.ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) )

Proof of Theorem hstel2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishst 23709 . . . 4  |-  ( S  e.  CHStates 
<->  ( S : CH --> ~H  /\  ( normh `  ( S `  ~H )
)  =  1  /\ 
A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) ) ) )
21simp3bi 974 . . 3  |-  ( S  e.  CHStates  ->  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) ) )
32ad2antrr 707 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  A. x  e.  CH  A. y  e.  CH  (
x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) ) )
4 sseq1 3361 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
5 fveq2 5720 . . . . . . . . . 10  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
65oveq1d 6088 . . . . . . . . 9  |-  ( x  =  A  ->  (
( S `  x
)  .ih  ( S `  y ) )  =  ( ( S `  A )  .ih  ( S `  y )
) )
76eqeq1d 2443 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  <->  (
( S `  A
)  .ih  ( S `  y ) )  =  0 ) )
8 oveq1 6080 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
98fveq2d 5724 . . . . . . . . 9  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
105oveq1d 6088 . . . . . . . . 9  |-  ( x  =  A  ->  (
( S `  x
)  +h  ( S `
 y ) )  =  ( ( S `
 A )  +h  ( S `  y
) ) )
119, 10eqeq12d 2449 . . . . . . . 8  |-  ( x  =  A  ->  (
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) )  <->  ( S `  ( A  vH  y
) )  =  ( ( S `  A
)  +h  ( S `
 y ) ) ) )
127, 11anbi12d 692 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( S `
 x )  .ih  ( S `  y ) )  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) )  <-> 
( ( ( S `
 A )  .ih  ( S `  y ) )  =  0  /\  ( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +h  ( S `  y
) ) ) ) )
134, 12imbi12d 312 . . . . . 6  |-  ( x  =  A  ->  (
( x  C_  ( _|_ `  y )  -> 
( ( ( S `
 x )  .ih  ( S `  y ) )  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) )  <->  ( A  C_  ( _|_ `  y )  ->  ( ( ( S `  A ) 
.ih  ( S `  y ) )  =  0  /\  ( S `
 ( A  vH  y ) )  =  ( ( S `  A )  +h  ( S `  y )
) ) ) ) )
14 fveq2 5720 . . . . . . . 8  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1514sseq2d 3368 . . . . . . 7  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
16 fveq2 5720 . . . . . . . . . 10  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1716oveq2d 6089 . . . . . . . . 9  |-  ( y  =  B  ->  (
( S `  A
)  .ih  ( S `  y ) )  =  ( ( S `  A )  .ih  ( S `  B )
) )
1817eqeq1d 2443 . . . . . . . 8  |-  ( y  =  B  ->  (
( ( S `  A )  .ih  ( S `  y )
)  =  0  <->  (
( S `  A
)  .ih  ( S `  B ) )  =  0 ) )
19 oveq2 6081 . . . . . . . . . 10  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
2019fveq2d 5724 . . . . . . . . 9  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
2116oveq2d 6089 . . . . . . . . 9  |-  ( y  =  B  ->  (
( S `  A
)  +h  ( S `
 y ) )  =  ( ( S `
 A )  +h  ( S `  B
) ) )
2220, 21eqeq12d 2449 . . . . . . . 8  |-  ( y  =  B  ->  (
( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +h  ( S `  y
) )  <->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +h  ( S `
 B ) ) ) )
2318, 22anbi12d 692 . . . . . . 7  |-  ( y  =  B  ->  (
( ( ( S `
 A )  .ih  ( S `  y ) )  =  0  /\  ( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +h  ( S `  y
) ) )  <->  ( (
( S `  A
)  .ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) ) )
2415, 23imbi12d 312 . . . . . 6  |-  ( y  =  B  ->  (
( A  C_  ( _|_ `  y )  -> 
( ( ( S `
 A )  .ih  ( S `  y ) )  =  0  /\  ( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +h  ( S `  y
) ) ) )  <-> 
( A  C_  ( _|_ `  B )  -> 
( ( ( S `
 A )  .ih  ( S `  B ) )  =  0  /\  ( S `  ( A  vH  B ) )  =  ( ( S `
 A )  +h  ( S `  B
) ) ) ) ) )
2513, 24rspc2v 3050 . . . . 5  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A. x  e. 
CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) )  ->  ( A  C_  ( _|_ `  B
)  ->  ( (
( S `  A
)  .ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) ) ) )
2625com23 74 . . . 4  |-  ( ( A  e.  CH  /\  B  e.  CH )  ->  ( A  C_  ( _|_ `  B )  -> 
( A. x  e. 
CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) )  ->  ( (
( S `  A
)  .ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) ) ) )
2726impr 603 . . 3  |-  ( ( A  e.  CH  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B
) ) )  -> 
( A. x  e. 
CH  A. y  e.  CH  ( x  C_  ( _|_ `  y )  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  /\  ( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) ) ) )  ->  ( (
( S `  A
)  .ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) ) )
2827adantll 695 . 2  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( A. x  e.  CH  A. y  e. 
CH  ( x  C_  ( _|_ `  y )  ->  ( ( ( S `  x ) 
.ih  ( S `  y ) )  =  0  /\  ( S `
 ( x  vH  y ) )  =  ( ( S `  x )  +h  ( S `  y )
) ) )  -> 
( ( ( S `
 A )  .ih  ( S `  B ) )  =  0  /\  ( S `  ( A  vH  B ) )  =  ( ( S `
 A )  +h  ( S `  B
) ) ) ) )
293, 28mpd 15 1  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( ( ( S `  A ) 
.ih  ( S `  B ) )  =  0  /\  ( S `
 ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983   ~Hchil 22414    +h cva 22415    .ih csp 22417   normhcno 22418   CHcch 22424   _|_cort 22425    vH chj 22428   CHStateschst 22458
This theorem is referenced by:  hstorth  23715  hstosum  23716
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-hilex 22494
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-sh 22701  df-ch 22716  df-hst 23707
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