Theorem hstelt 10098

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 1814	. 2 |- (S e. CHStates -> S e. V)
2		chex 9050	. . . 4 |- CH e. V
3		fex 3647	. . . 4 |- ((S:CH-->H~ /\ CH e. V) -> S e. V)
4	2, 3	mpan2 695	. . 3 |- (S:CH-->H~ -> S e. V)
5	4	3ad2ant1 799	. 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 3616	. . . 4 |- (f = S -> (f:CH-->H~ <-> S:CH-->H~))
7		fveq1 3718	. . . . . 6 |- (f = S -> (f` H~) = (S` H~))
8	7	fveq2d 3723	. . . . 5 |- (f = S -> (normh` (f` H~)) = (normh` (S` H~)))
9	8	eqeq1d 1481	. . . 4 |- (f = S -> ((normh` (f` H~)) = 1 <-> (normh` (S` H~)) = 1))
10		fveq1 3718	. . . . . . . . 9 |- (f = S -> (f` x) = (S` x))
11		fveq1 3718	. . . . . . . . 9 |- (f = S -> (f` y) = (S` y))
12	10, 11	opreq12d 3973	. . . . . . . 8 |- (f = S -> ((f` x) .ih (f` y)) = ((S` x) .ih (S` y)))
13	12	eqeq1d 1481	. . . . . . 7 |- (f = S -> (((f` x) .ih (f` y)) = 0 <-> ((S` x) .ih (S` y)) = 0))
14		fveq1 3718	. . . . . . . 8 |- (f = S -> (f` (x vH y)) = (S` (x vH y)))
15	10, 11	opreq12d 3973	. . . . . . . 8 |- (f = S -> ((f` x) +h (f` y)) = ((S` x) +h (S` y)))
16	14, 15	eqeq12d 1487	. . . . . . 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 627	. . . . . 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 611	. . . . 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 1678	. . . 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 892	. . 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 10096	. . 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 1897	. 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 678	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 774   = wceq 955   e. wcel 957  A.wral 1643  Vcvv 1808   (_ wss 2044  -->wf 3174  ` cfv 3178  (class class class)co 3958  0cc0 5217  1c1 5218  H~chil 8743   +h cva 8744   .ih csp 8748  normhcno 8749  CHcch 8753  _|_cort 8754   vH chj 8757  CHStateschst 8787

This theorem is referenced by:  hstclt 10100  hst1t 10101  hstel2t 10102  hstrlem3a 10143

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-hilex 8824

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194  df-opr 3960  df-sh 9031  df-ch 9047  df-hst 10096

Copyright terms: Public domain