Theorem spanval 29113

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. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
spanval	⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem spanval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-span 29089	. 2 ⊢ span = (𝑦 ∈ 𝒫 ℋ ↦ ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥})
2		sseq1 3995	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3	2	rabbidv 3483	. . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥})
4	3	inteqd 4884	. 2 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥})
5		ax-hilex 28779	. . . 4 ⊢ ℋ ∈ V
6	5	elpw2 5251	. . 3 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ)
7	6	biimpri 230	. 2 ⊢ (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ)
8		helsh 29025	. . . 4 ⊢ ℋ ∈ Sℋ
9		sseq2 3996	. . . . 5 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
10	9	rspcev 3626	. . . 4 ⊢ (( ℋ ∈ Sℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥)
11	8, 10	mpan 688	. . 3 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥)
12		intexrab 5246	. . 3 ⊢ (∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V)
13	11, 12	sylib 220	. 2 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V)
14	1, 4, 7, 13	fvmptd3 6794	1 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥})

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  ∃wrex 3142  {crab 3145  Vcvv 3497   ⊆ wss 3939  𝒫 cpw 4542  ∩ cint 4879  ‘cfv 6358   ℋchba 28699   S_ℋ csh 28708  spancspn 28712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-1cn 10598  ax-addcl 10600  ax-hilex 28779  ax-hfvadd 28780  ax-hv0cl 28783  ax-hfvmul 28785

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-map 8411  df-nn 11642  df-hlim 28752  df-sh 28987  df-ch 29001  df-span 29089

This theorem is referenced by:  spancl  29116  spanss2  29125  spanid  29127  spanss  29128  shsval3i  29168  elspani  29323