Theorem hstel2 11791

Description: Properties of a Hilbert-space-valued state.

Assertion

Ref	Expression
hstel2	|- (((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))))

Proof of Theorem hstel2

Step	Hyp	Ref	Expression
1		hstel 11787	. . . 4 |- (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))))))
2	1	simp3bi 893	. . 3 |- (S e. CHStates -> 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)))))
3	2	ad2antrr 440	. 2 |- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A C_ (_|_` B))) -> 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)))))
4		sseq1 2637	. . . . . . 7 |- (x = A -> (x C_ (_|_` y) <-> A C_ (_|_` y)))
5		fveq2 4681	. . . . . . . . . 10 |- (x = A -> (S` x) = (S` A))
6	5	opreq1d 4897	. . . . . . . . 9 |- (x = A -> ((S` x) .ih (S` y)) = ((S` A) .ih (S` y)))
7	6	eqeq1d 1892	. . . . . . . 8 |- (x = A -> (((S` x) .ih (S` y)) = 0 <-> ((S` A) .ih (S` y)) = 0))
8		opreq1 4889	. . . . . . . . . 10 |- (x = A -> (x vH y) = (A vH y))
9	8	fveq2d 4685	. . . . . . . . 9 |- (x = A -> (S` (x vH y)) = (S` (A vH y)))
10	5	opreq1d 4897	. . . . . . . . 9 |- (x = A -> ((S` x) +h (S` y)) = ((S` A) +h (S` y)))
11	9, 10	eqeq12d 1899	. . . . . . . 8 |- (x = A -> ((S` (x vH y)) = ((S` x) +h (S` y)) <-> (S` (A vH y)) = ((S` A) +h (S` y))))
12	7, 11	anbi12d 690	. . . . . . 7 |- (x = A -> ((((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y))) <-> (((S` A) .ih (S` y)) = 0 /\ (S` (A vH y)) = ((S` A) +h (S` y)))))
13	4, 12	imbi12d 688	. . . . . 6 |- (x = A -> ((x C_ (_|_` y) -> (((S` x) .ih (S` y)) = 0 /\ (S` (x vH y)) = ((S` x) +h (S` y)))) <-> (A C_ (_|_` y) -> (((S` A) .ih (S` y)) = 0 /\ (S` (A vH y)) = ((S` A) +h (S` y))))))
14		fveq2 4681	. . . . . . . 8 |- (y = B -> (_|_` y) = (_|_` B))
15	14	sseq2d 2645	. . . . . . 7 |- (y = B -> (A C_ (_|_` y) <-> A C_ (_|_` B)))
16		fveq2 4681	. . . . . . . . . 10 |- (y = B -> (S` y) = (S` B))
17	16	opreq2d 4898	. . . . . . . . 9 |- (y = B -> ((S` A) .ih (S` y)) = ((S` A) .ih (S` B)))
18	17	eqeq1d 1892	. . . . . . . 8 |- (y = B -> (((S` A) .ih (S` y)) = 0 <-> ((S` A) .ih (S` B)) = 0))
19		opreq2 4890	. . . . . . . . . 10 |- (y = B -> (A vH y) = (A vH B))
20	19	fveq2d 4685	. . . . . . . . 9 |- (y = B -> (S` (A vH y)) = (S` (A vH B)))
21	16	opreq2d 4898	. . . . . . . . 9 |- (y = B -> ((S` A) +h (S` y)) = ((S` A) +h (S` B)))
22	20, 21	eqeq12d 1899	. . . . . . . 8 |- (y = B -> ((S` (A vH y)) = ((S` A) +h (S` y)) <-> (S` (A vH B)) = ((S` A) +h (S` B))))
23	18, 22	anbi12d 690	. . . . . . 7 |- (y = B -> ((((S` A) .ih (S` y)) = 0 /\ (S` (A vH y)) = ((S` A) +h (S` y))) <-> (((S` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B)))))
24	15, 23	imbi12d 688	. . . . . 6 |- (y = B -> ((A C_ (_|_` y) -> (((S` A) .ih (S` y)) = 0 /\ (S` (A vH y)) = ((S` A) +h (S` y)))) <-> (A C_ (_|_` B) -> (((S` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B))))))
25	13, 24	rcla42v 2384	. . . . 5 |- ((A e. CH /\ B e. CH) -> (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)))) -> (A C_ (_|_` B) -> (((S` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B))))))
26	25	com23 36	. . . 4 |- ((A e. CH /\ B e. CH) -> (A C_ (_|_` B) -> (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` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B))))))
27	26	impr 422	. . 3 |- ((A e. CH /\ (B e. CH /\ A C_ (_|_` B))) -> (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` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B)))))
28	27	adantll 428	. 2 |- (((S e. CHStates /\ A e. CH) /\ (B e. CH /\ A C_ (_|_` B))) -> (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` A) .ih (S` B)) = 0 /\ (S` (A vH B)) = ((S` A) +h (S` B)))))
29	3, 28	mpd 29	1 |- (((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))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   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:  hstorth 11792  hstosum 11793

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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