Theorem hstel 11787

Description: Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9.

Assertion

Ref	Expression
hstel	|- (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))))))

Distinct variable group:   x,y,S

Proof of Theorem hstel

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (S e. CHStates -> S e. _V)
2		chex 10728	. . . 4 |- CH e. _V
3		fex 4595	. . . 4 |- ((S:CH-->~H /\ CH e. _V) -> S e. _V)
4	2, 3	mpan2 760	. . 3 |- (S:CH-->~H -> S e. _V)
5	4	3ad2ant1 897	. 2 |- ((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))))) -> S e. _V)
6		feq1 4551	. . . 4 |- (f = S -> (f:CH-->~H <-> S:CH-->~H))
7		fveq1 4680	. . . . . 6 |- (f = S -> (f` ~H) = (S` ~H))
8	7	fveq2d 4685	. . . . 5 |- (f = S -> (normh` (f` ~H)) = (normh` (S` ~H)))
9	8	eqeq1d 1892	. . . 4 |- (f = S -> ((normh` (f` ~H)) = 1 <-> (normh` (S` ~H)) = 1))
10		fveq1 4680	. . . . . . . . 9 |- (f = S -> (f` x) = (S` x))
11		fveq1 4680	. . . . . . . . 9 |- (f = S -> (f` y) = (S` y))
12	10, 11	opreq12d 4900	. . . . . . . 8 |- (f = S -> ((f` x) .ih (f` y)) = ((S` x) .ih (S` y)))
13	12	eqeq1d 1892	. . . . . . 7 |- (f = S -> (((f` x) .ih (f` y)) = 0 <-> ((S` x) .ih (S` y)) = 0))
14		fveq1 4680	. . . . . . . 8 |- (f = S -> (f` (x vH y)) = (S` (x vH y)))
15	10, 11	opreq12d 4900	. . . . . . . 8 |- (f = S -> ((f` x) +h (f` y)) = ((S` x) +h (S` y)))
16	14, 15	eqeq12d 1899	. . . . . . 7 |- (f = S -> ((f` (x vH y)) = ((f` x) +h (f` y)) <-> (S` (x vH y)) = ((S` x) +h (S` y))))
17	13, 16	anbi12d 690	. . . . . 6 |- (f = S -> ((((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y))) <-> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y)))))
18	17	imbi2d 674	. . . . 5 |- (f = S -> ((x C_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))) <-> (x C_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))))
19	18	2ralbidv 2140	. . . 4 |- (f = S -> (A.x e. CH A.y e. CH (x C_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))) <-> 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))))))
20	6, 9, 19	3anbi123d 1168	. . 3 |- (f = S -> ((f:CH-->~H /\ (normh` (f` ~H)) = 1 /\ A.x e. CH A.y e. CH (x C_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y))))) <-> (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)))))))
21		df-hst 11785	. . 3 |- CHStates = {f | (f:CH-->~H /\ (normh` (f` ~H)) = 1 /\ A.x e. CH A.y e. CH (x C_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))))}
22	20, 21	elab2g 2406	. 2 |- (S e. _V -> (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)))))))
23	1, 5, 22	pm5.21nii 743	1 |- (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))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  -->wf 3994  ` cfv 3998  (class class class)co 4884  0cc0 6386  1c1 6387  ~Hchil 10420   +h cva 10421   .ih csp 10425  normhcno 10426  CHcch 10430  _|_cort 10431   vH chj 10434  CHStateschst 10464

This theorem is referenced by:  hstcl 11789  hst1a 11790  hstel2 11791  hstrlem3a 11832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-sh 10709  df-ch 10725  df-hst 11785

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