Theorem hstelt 10052

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

Assertion

Ref	Expression
hstelt	|- (S e. CHStates <-> (S:CH-->H~ /\ (normh` (S` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` 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 hstelt

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (S e. CHStates -> S e. V)
2		chex 9016	. . . 4 |- CH e. V
3		fex 3637	. . . 4 |- ((S:CH-->H~ /\ CH e. V) -> S e. V)
4	2, 3	mpan2 694	. . 3 |- (S:CH-->H~ -> S e. V)
5	4	3ad2ant1 798	. 2 |- ((S:CH-->H~ /\ (normh` (S` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))) -> S e. V)
6		feq1 3606	. . . 4 |- (f = S -> (f:CH-->H~ <-> S:CH-->H~))
7		fveq1 3708	. . . . . 6 |- (f = S -> (f` H~) = (S` H~))
8	7	fveq2d 3713	. . . . 5 |- (f = S -> (normh` (f` H~)) = (normh` (S` H~)))
9	8	eqeq1d 1475	. . . 4 |- (f = S -> ((normh` (f` H~)) = 1 <-> (normh` (S` H~)) = 1))
10		fveq1 3708	. . . . . . . . 9 |- (f = S -> (f` x) = (S` x))
11		fveq1 3708	. . . . . . . . 9 |- (f = S -> (f` y) = (S` y))
12	10, 11	opreq12d 3963	. . . . . . . 8 |- (f = S -> ((f` x) .ih (f` y)) = ((S` x) .ih (S` y)))
13	12	eqeq1d 1475	. . . . . . 7 |- (f = S -> (((f` x) .ih (f` y)) = 0 <-> ((S` x) .ih (S` y)) = 0))
14		fveq1 3708	. . . . . . . 8 |- (f = S -> (f` (x vH y)) = (S` (x vH y)))
15	10, 11	opreq12d 3963	. . . . . . . 8 |- (f = S -> ((f` x) +h (f` y)) = ((S` x) +h (S` y)))
16	14, 15	eqeq12d 1481	. . . . . . 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 626	. . . . . 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 610	. . . . 5 |- (f = S -> ((x (_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))) <-> (x (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))))
19	18	2ralbidv 1672	. . . 4 |- (f = S -> (A.x e. CH A.y e. CH (x (_ (_|_` 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 (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))))
20	6, 9, 19	3anbi123d 890	. . 3 |- (f = S -> ((f:CH-->H~ /\ (normh` (f` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` 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 (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y)))))))
21		df-hst 10050	. . 3 |- CHStates = {f | (f:CH-->H~ /\ (normh` (f` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((f` x) .ih (f` y)) = 0 /\ (f` (x vH y)) = ((f` x) +h (f` y)))))}
22	20, 21	elab2g 1891	. 2 |- (S e. V -> (S e. CHStates <-> (S:CH-->H~ /\ (normh` (S` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y)))))))
23	1, 5, 22	pm5.21nii 677	1 |- (S e. CHStates <-> (S:CH-->H~ /\ (normh` (S` H~)) = 1 /\ A.x e. CH A.y e. CH (x (_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   (_ wss 2037  -->wf 3168  ` cfv 3172  (class class class)co 3948  0cc0 5206  1c1 5207  H~chil 8727   +h cva 8728   .ih csp 8732  normhcno 8733  CHcch 8737  _|_cort 8738   vH chj 8741  CHStateschst 8771

This theorem is referenced by:  hstclt 10054  hst1t 10055  hstel2t 10056  hstrlem3a 10097

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-sh 8997  df-ch 9013  df-hst 10050

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