Theorem spanvalt 9299

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:   x,A

Proof of Theorem spanvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8869	. . 3 |- H~ e. V
2	1	elpw2 2728	. 2 |- (A e. P~H~ <-> A (_ H~)
3		helsh 9117	. . . . . 6 |- H~ e. SH
4		sseq2 2083	. . . . . . 7 |- (x = H~ -> (A (_ x <-> A (_ H~))
5	4	rcla4ev 1877	. . . . . 6 |- ((H~ e. SH /\ A (_ H~) -> E.x e. SH A (_ x)
6	3, 5	mpan 695	. . . . 5 |- (A (_ H~ -> E.x e. SH A (_ x)
7	2, 6	sylbi 199	. . . 4 |- (A e. P~H~ -> E.x e. SH A (_ x)
8		intexrab 2732	. . . 4 |- (E.x e. SH A (_ x <-> |^|{x e. SH | A (_ x} e. V)
9	7, 8	sylib 198	. . 3 |- (A e. P~H~ -> |^|{x e. SH | A (_ x} e. V)
10		sseq1 2082	. . . . . 6 |- (y = A -> (y (_ x <-> A (_ x))
11	10	rabbisdv 1807	. . . . 5 |- (y = A -> {x e. SH | y (_ x} = {x e. SH | A (_ x})
12	11	inteqd 2538	. . . 4 |- (y = A -> |^|{x e. SH | y (_ x} = |^|{x e. SH | A (_ x})
13		df-span 9274	. . . . 5 |- span = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
14	1	elpw2 2728	. . . . . . 7 |- (y e. P~H~ <-> y (_ H~)
15	14	anbi1i 481	. . . . . 6 |- ((y e. P~H~ /\ z = |^|{x e. SH | y (_ x}) <-> (y (_ H~ /\ z = |^|{x e. SH | y (_ x}))
16	15	opabbii 2671	. . . . 5 |- {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})} = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
17	13, 16	eqtr4 1498	. . . 4 |- span = {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})}
18	12, 17	fvopab4g 3779	. . 3 |- ((A e. P~H~ /\ |^|{x e. SH | A (_ x} e. V) -> (span` A) = |^|{x e. SH | A (_ x})
19	9, 18	mpdan 704	. 2 |- (A e. P~H~ -> (span` A) = |^|{x e. SH | A (_ x})
20	2, 19	sylbir 201	1 |- (A (_ H~ -> (span` A) = |^|{x e. SH | A (_ x})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646  {crab 1648  Vcvv 1811   (_ wss 2047  P~cpw 2401  |^|cint 2533  {copab 2666  ` cfv 3182  H~chil 8788  SHcsh 8797  spancspn 8801

This theorem is referenced by:  spanclt 9304  spanss2 9314  spanid 9317  spanss 9318  shsumval3 9361  elspan 9466

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-9 965  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-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869  ax-hfvadd 8870  ax-hv0cl 8873  ax-hfvmul 8875

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1652  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-int 2534  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-hlim 8841  df-sh 9076  df-ch 9092  df-span 9274

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