Theorem hstorth 29079

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

Step	Hyp	Ref	Expression
1		hstel2 29078	. 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 ) ) ) )
2	1	simpld 475	1 |- ( ( ( S e. CHStates /\ A e. CH ) /\ ( B e. CH /\ A C_ ( _|_ ` B ) ) ) -> ( ( S ` A ) .ih ( S ` B ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ` cfv 5888  (class class class)co 6650   0cc0 9936   +h cva 27777   .ih csp 27779   CHcch 27786   _|_cort 27787   vH chj 27790   CHStateschst 27820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-sh 28064  df-ch 28078  df-hst 29071

This theorem is referenced by:  hstnmoc  29082  hstpyth  29088  hstoh  29091  hst0  29092