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Theorem hstosum 23677
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 23675 . 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 450 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 359    = wceq 1649    e. wcel 1721    C_ wss 3280   ` cfv 5413  (class class class)co 6040   0cc0 8946    +h cva 22376    .ih csp 22378   CHcch 22385   _|_cort 22386    vH chj 22389   CHStateschst 22419
This theorem is referenced by:  hstoc  23678  hstpyth  23685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-hilex 22455
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979  df-sh 22662  df-ch 22677  df-hst 23668
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