Theorem hstelt 10142

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 1817	. 2 |- (S e. CHStates -> S e. V)
2		chex 9095	. . . 4 |- CH e. V
3		fex 3652	. . . 4 |- ((S:CH-->H~ /\ CH e. V) -> S e. V)
4	2, 3	mpan2 696	. . 3 |- (S:CH-->H~ -> S e. V)
5	4	3ad2ant1 800	. 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 3620	. . . 4 |- (f = S -> (f:CH-->H~ <-> S:CH-->H~))
7		fveq1 3723	. . . . . 6 |- (f = S -> (f` H~) = (S` H~))
8	7	fveq2d 3728	. . . . 5 |- (f = S -> (normh` (f` H~)) = (normh` (S` H~)))
9	8	eqeq1d 1483	. . . 4 |- (f = S -> ((normh` (f` H~)) = 1 <-> (normh` (S` H~)) = 1))
10		fveq1 3723	. . . . . . . . 9 |- (f = S -> (f` x) = (S` x))
11		fveq1 3723	. . . . . . . . 9 |- (f = S -> (f` y) = (S` y))
12	10, 11	opreq12d 3978	. . . . . . . 8 |- (f = S -> ((f` x) .ih (f` y)) = ((S` x) .ih (S` y)))
13	12	eqeq1d 1483	. . . . . . 7 |- (f = S -> (((f` x) .ih (f` y)) = 0 <-> ((S` x) .ih (S` y)) = 0))
14		fveq1 3723	. . . . . . . 8 |- (f = S -> (f` (x vH y)) = (S` (x vH y)))
15	10, 11	opreq12d 3978	. . . . . . . 8 |- (f = S -> ((f` x) +h (f` y)) = ((S` x) +h (S` y)))
16	14, 15	eqeq12d 1489	. . . . . . 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 628	. . . . . 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 612	. . . . 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 1680	. . . 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 893	. . 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 10140	. . 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 1900	. 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 679	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 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  -->wf 3178  ` cfv 3182  (class class class)co 3963  0cc0 5234  1c1 5235  H~chil 8788   +h cva 8789   .ih csp 8793  normhcno 8794  CHcch 8798  _|_cort 8799   vH chj 8802  CHStateschst 8832

This theorem is referenced by:  hstclt 10144  hst1t 10145  hstel2t 10146  hstrlem3a 10187

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-sh 9076  df-ch 9092  df-hst 10140

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