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Mirrors > Home > HSE Home > Th. List > spanval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
spanval | ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-span 30995 | . 2 ⊢ span = (𝑦 ∈ 𝒫 ℋ ↦ ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥}) | |
2 | sseq1 4007 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | rabbidv 3439 | . . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
4 | 3 | inteqd 4955 | . 2 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
5 | ax-hilex 30685 | . . . 4 ⊢ ℋ ∈ V | |
6 | 5 | elpw2 5345 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
7 | 6 | biimpri 227 | . 2 ⊢ (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ) |
8 | helsh 30931 | . . . 4 ⊢ ℋ ∈ Sℋ | |
9 | sseq2 4008 | . . . . 5 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ)) | |
10 | 9 | rspcev 3612 | . . . 4 ⊢ (( ℋ ∈ Sℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
11 | 8, 10 | mpan 687 | . . 3 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
12 | intexrab 5340 | . . 3 ⊢ (∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V) | |
13 | 11, 12 | sylib 217 | . 2 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V) |
14 | 1, 4, 7, 13 | fvmptd3 7021 | 1 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 {crab 3431 Vcvv 3473 ⊆ wss 3948 𝒫 cpw 4602 ∩ cint 4950 ‘cfv 6543 ℋchba 30605 Sℋ csh 30614 spancspn 30618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-1cn 11174 ax-addcl 11176 ax-hilex 30685 ax-hfvadd 30686 ax-hv0cl 30689 ax-hfvmul 30691 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-map 8828 df-nn 12220 df-hlim 30658 df-sh 30893 df-ch 30907 df-span 30995 |
This theorem is referenced by: spancl 31022 spanss2 31031 spanid 31033 spanss 31034 shsval3i 31074 elspani 31229 |
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