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Theorem hstel2 22799
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 22794 . . . 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 972 . . 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 706 . 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 3199 . . . . . . 7  |-  ( x  =  A  ->  (
x  C_  ( _|_ `  y )  <->  A  C_  ( _|_ `  y ) ) )
5 fveq2 5525 . . . . . . . . . 10  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
65oveq1d 5873 . . . . . . . . 9  |-  ( x  =  A  ->  (
( S `  x
)  .ih  ( S `  y ) )  =  ( ( S `  A )  .ih  ( S `  y )
) )
76eqeq1d 2291 . . . . . . . 8  |-  ( x  =  A  ->  (
( ( S `  x )  .ih  ( S `  y )
)  =  0  <->  (
( S `  A
)  .ih  ( S `  y ) )  =  0 ) )
8 oveq1 5865 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x  vH  y )  =  ( A  vH  y ) )
98fveq2d 5529 . . . . . . . . 9  |-  ( x  =  A  ->  ( S `  ( x  vH  y ) )  =  ( S `  ( A  vH  y ) ) )
105oveq1d 5873 . . . . . . . . 9  |-  ( x  =  A  ->  (
( S `  x
)  +h  ( S `
 y ) )  =  ( ( S `
 A )  +h  ( S `  y
) ) )
119, 10eqeq12d 2297 . . . . . . . 8  |-  ( x  =  A  ->  (
( S `  (
x  vH  y )
)  =  ( ( S `  x )  +h  ( S `  y ) )  <->  ( S `  ( A  vH  y
) )  =  ( ( S `  A
)  +h  ( S `
 y ) ) ) )
127, 11anbi12d 691 . . . . . . 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 311 . . . . . 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 5525 . . . . . . . 8  |-  ( y  =  B  ->  ( _|_ `  y )  =  ( _|_ `  B
) )
1514sseq2d 3206 . . . . . . 7  |-  ( y  =  B  ->  ( A  C_  ( _|_ `  y
)  <->  A  C_  ( _|_ `  B ) ) )
16 fveq2 5525 . . . . . . . . . 10  |-  ( y  =  B  ->  ( S `  y )  =  ( S `  B ) )
1716oveq2d 5874 . . . . . . . . 9  |-  ( y  =  B  ->  (
( S `  A
)  .ih  ( S `  y ) )  =  ( ( S `  A )  .ih  ( S `  B )
) )
1817eqeq1d 2291 . . . . . . . 8  |-  ( y  =  B  ->  (
( ( S `  A )  .ih  ( S `  y )
)  =  0  <->  (
( S `  A
)  .ih  ( S `  B ) )  =  0 ) )
19 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  B  ->  ( A  vH  y )  =  ( A  vH  B
) )
2019fveq2d 5529 . . . . . . . . 9  |-  ( y  =  B  ->  ( S `  ( A  vH  y ) )  =  ( S `  ( A  vH  B ) ) )
2116oveq2d 5874 . . . . . . . . 9  |-  ( y  =  B  ->  (
( S `  A
)  +h  ( S `
 y ) )  =  ( ( S `
 A )  +h  ( S `  B
) ) )
2220, 21eqeq12d 2297 . . . . . . . 8  |-  ( y  =  B  ->  (
( S `  ( A  vH  y ) )  =  ( ( S `
 A )  +h  ( S `  y
) )  <->  ( S `  ( A  vH  B
) )  =  ( ( S `  A
)  +h  ( S `
 B ) ) ) )
2318, 22anbi12d 691 . . . . . . 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 311 . . . . . 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 2890 . . . . 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 72 . . . 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 602 . . 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 694 . 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 14 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 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   ~Hchil 21499    +h cva 21500    .ih csp 21502   normhcno 21503   CHcch 21509   _|_cort 21510    vH chj 21513   CHStateschst 21543
This theorem is referenced by:  hstorth  22800  hstosum  22801
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-hilex 21579
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-eu 2147  df-mo 2148  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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-sh 21786  df-ch 21801  df-hst 22792
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