Theorem spanvalt 5300

Description: Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276.

Assertion

Ref	Expression
spanvalt	|- (A (_ H~ -> (span` A) = |^|{x e. SH | A (_ x})

Distinct variable group(s):   x,A

Proof of Theorem spanvalt

Step	Hyp	Ref	Expression
1		ax-hilex 4983	. . 3 |- H~ e. V
2		elpw2g 1803	. . 3 |- (H~ e. V -> (A e. P~H~ <-> A (_ H~))
3	1, 2	ax-mp 6	. 2 |- (A e. P~H~ <-> A (_ H~)
4		helsh 5152	. . . . . 6 |- H~ e. SH
5		sseq2 1522	. . . . . . 7 |- (x = H~ -> (A (_ x <-> A (_ H~))
6	5	rcla4ev 1403	. . . . . 6 |- ((H~ e. SH /\ A (_ H~) -> E.x e. SH A (_ x)
7	4, 6	mpan 518	. . . . 5 |- (A (_ H~ -> E.x e. SH A (_ x)
8	3, 7	sylbi 174	. . . 4 |- (A e. P~H~ -> E.x e. SH A (_ x)
9		intexrab 1988	. . . 4 |- (E.x e. SH A (_ x <-> |^|{x e. SH | A (_ x} e. V)
10	8, 9	sylib 173	. . 3 |- (A e. P~H~ -> |^|{x e. SH | A (_ x} e. V)
11		sseq1 1521	. . . . . 6 |- (y = A -> (y (_ x <-> A (_ x))
12	11	birabsdv 1344	. . . . 5 |- (y = A -> {x e. SH | y (_ x} = {x e. SH | A (_ x})
13	12	inteqd 1970	. . . 4 |- (y = A -> |^|{x e. SH | y (_ x} = |^|{x e. SH | A (_ x})
14		df-span 5276	. . . . 5 |- span = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
15		visset 1350	. . . . . . . 8 |- y e. V
16	15	elpw 1801	. . . . . . 7 |- (y e. P~H~ <-> y (_ H~)
17	16	anbi1i 368	. . . . . 6 |- ((y e. P~H~ /\ z = |^|{x e. SH | y (_ x}) <-> (y (_ H~ /\ z = |^|{x e. SH | y (_ x}))
18	17	biopabi 2103	. . . . 5 |- {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})} = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
19	14, 18	eqtr4 1122	. . . 4 |- span = {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})}
20	13, 19	fvopab4g 2870	. . 3 |- ((A e. P~H~ /\ |^|{x e. SH | A (_ x} e. V) -> (span` A) = |^|{x e. SH | A (_ x})
21	10, 20	mpdan 527	. 2 |- (A e. P~H~ -> (span` A) = |^|{x e. SH | A (_ x})
22	3, 21	sylbir 176	1 |- (A (_ H~ -> (span` A) = |^|{x e. SH | A (_ x})

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204  Vcvv 1348   (_ wss 1487  P~cpw 1798  |^|cint 1965  {copab 2055  ` cfv 2422  H~chil 4958  SHcsh 4967  spancspn 4971

This theorem is referenced by:  spanclt 5305  spanss2 5315  spanid 5318  spanss 5319  shsumval3 5362  elspan 5449

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983  ax-hvaddcl 4984  ax-hvzercl 4987  ax-hvmulcl 4989

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-hlim 5107  df-sh 5114  df-ch 5127  df-span 5276

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