Theorem spanvalt 9237

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 8808	. . 3 |- H~ e. V
2	1	elpw2 2723	. 2 |- (A e. P~H~ <-> A (_ H~)
3		helsh 9056	. . . . . 6 |- H~ e. SH
4		sseq2 2079	. . . . . . 7 |- (x = H~ -> (A (_ x <-> A (_ H~))
5	4	rcla4ev 1873	. . . . . 6 |- ((H~ e. SH /\ A (_ H~) -> E.x e. SH A (_ x)
6	3, 5	mpan 694	. . . . 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 2727	. . . 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 2078	. . . . . 6 |- (y = A -> (y (_ x <-> A (_ x))
11	10	rabbisdv 1803	. . . . 5 |- (y = A -> {x e. SH | y (_ x} = {x e. SH | A (_ x})
12	11	inteqd 2533	. . . 4 |- (y = A -> |^|{x e. SH | y (_ x} = |^|{x e. SH | A (_ x})
13		df-span 9212	. . . . 5 |- span = {<.y, z>. | (y (_ H~ /\ z = |^|{x e. SH | y (_ x})}
14	1	elpw2 2723	. . . . . . 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 2666	. . . . 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 1495	. . . 4 |- span = {<.y, z>. | (y e. P~H~ /\ z = |^|{x e. SH | y (_ x})}
18	12, 17	fvopab4g 3770	. . 3 |- ((A e. P~H~ /\ |^|{x e. SH | A (_ x} e. V) -> (span` A) = |^|{x e. SH | A (_ x})
19	9, 18	mpdan 703	. 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 954   e. wcel 956  E.wrex 1643  {crab 1645  Vcvv 1807   (_ wss 2043  P~cpw 2397  |^|cint 2528  {copab 2661  ` cfv 3177  H~chil 8727  SHcsh 8736  spancspn 8740

This theorem is referenced by:  spanclt 9242  spanss2 9252  spanid 9255  spanss 9256  shsumval3 9299  elspan 9404

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-hilex 8808  ax-hfvadd 8809  ax-hv0cl 8812  ax-hfvmul 8814

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-hlim 8780  df-sh 9015  df-ch 9031  df-span 9212

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