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Theorem hstorth 26801
Description: Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hstorth  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( ( S `
 A )  .ih  ( S `  B ) )  =  0 )

Proof of Theorem hstorth
StepHypRef Expression
1 hstel2 26800 . 2  |-  ( ( ( 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 )
) ) )
21simpld 459 1  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( ( S `
 A )  .ih  ( S `  B ) )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3469   ` cfv 5579  (class class class)co 6275   0cc0 9481    +h cva 25499    .ih csp 25501   CHcch 25508   _|_cort 25509    vH chj 25512   CHStateschst 25542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-hilex 25578
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-sh 25786  df-ch 25801  df-hst 26793
This theorem is referenced by:  hstnmoc  26804  hstpyth  26810  hstoh  26813  hst0  26814
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