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Theorem hstosum 22915
Description: Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hstosum  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( S `  ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B ) ) )

Proof of Theorem hstosum
StepHypRef Expression
1 hstel2 22913 . 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 )
) ) )
21simprd 449 1  |-  ( ( ( S  e.  CHStates  /\  A  e.  CH )  /\  ( B  e.  CH  /\  A  C_  ( _|_ `  B ) ) )  ->  ( S `  ( A  vH  B ) )  =  ( ( S `  A )  +h  ( S `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228   ` cfv 5337  (class class class)co 5945   0cc0 8827    +h cva 21614    .ih csp 21616   CHcch 21623   _|_cort 21624    vH chj 21627   CHStateschst 21657
This theorem is referenced by:  hstoc  22916  hstpyth  22923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-hilex 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-map 6862  df-sh 21900  df-ch 21915  df-hst 22906
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