Theorem spanvalt 9214

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 8790	. . 3 |- H~ e. V
2	1	elpw2 2718	. 2 |- (A e. P~H~ <-> A (_ H~)
3		helsh 9038	. . . . . 6 |- H~ e. SH
4		sseq2 2073	. . . . . . 7 |- (x = H~ -> (A (_ x <-> A (_ H~))
5	4	rcla4ev 1868	. . . . . 6 |- ((H~ e. SH /\ A (_ H~) -> E.x e. SH A (_ x)
6	3, 5	mpan 693	. . . . 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 2722	. . . 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 2072	. . . . . 6 |- (y = A -> (y (_ x <-> A (_ x))
11	10	rabbisdv 1798	. . . . 5 |- (y = A -> {x e. SH | y (_ x} = {x e. SH | A (_ x})
12	11	inteqd 2528	. . . 4 |- (y = A -> |^|{x e. SH | y (_ x} = |^|{x e. SH | A (_ x})
13		df-span 9189	. . . . 5 |- span = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
14	1	elpw2 2718	. . . . . . 7 |- (y e. P~H~ <-> y (_ H~)
15	14	anbi1i 480	. . . . . 6 |- ((y e. P~H~ /\ z = |^|{x e. SH | y (_ x}) <-> (y (_ H~ /\ z = |^|{x e. SH | y (_ x}))
16	15	opabbii 2661	. . . . 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 1490	. . . 4 |- span = {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})}
18	12, 17	fvopab4g 3764	. . 3 |- ((A e. P~H~ /\ |^|{x e. SH | A (_ x} e. V) -> (span` A) = |^|{x e. SH | A (_ x})
19	9, 18	mpdan 702	. 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 953   e. wcel 955  E.wrex 1638  {crab 1640  Vcvv 1802   (_ wss 2037  P~cpw 2391  |^|cint 2523  {copab 2656  ` cfv 3172  H~chil 8727  SHcsh 8736  spancspn 8740

This theorem is referenced by:  spanclt 9219  spanss2 9229  spanid 9232  spanss 9233  shsumval3 9276  elspan 9381

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790  ax-hfvadd 8791  ax-hv0cl 8794  ax-hfvmul 8796

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-hlim 8780  df-sh 8997  df-ch 9013  df-span 9189

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