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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 29092 | . 2 ⊢ span = (𝑦 ∈ 𝒫 ℋ ↦ ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥}) | |
2 | sseq1 3940 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | rabbidv 3427 | . . 3 ⊢ (𝑦 = 𝐴 → {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
4 | 3 | inteqd 4843 | . 2 ⊢ (𝑦 = 𝐴 → ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
5 | ax-hilex 28782 | . . . 4 ⊢ ℋ ∈ V | |
6 | 5 | elpw2 5212 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
7 | 6 | biimpri 231 | . 2 ⊢ (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ) |
8 | helsh 29028 | . . . 4 ⊢ ℋ ∈ Sℋ | |
9 | sseq2 3941 | . . . . 5 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ)) | |
10 | 9 | rspcev 3571 | . . . 4 ⊢ (( ℋ ∈ Sℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
11 | 8, 10 | mpan 689 | . . 3 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
12 | intexrab 5207 | . . 3 ⊢ (∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V) | |
13 | 11, 12 | sylib 221 | . 2 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} ∈ V) |
14 | 1, 4, 7, 13 | fvmptd3 6768 | 1 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∩ cint 4838 ‘cfv 6324 ℋchba 28702 Sℋ csh 28711 spancspn 28715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-1cn 10584 ax-addcl 10586 ax-hilex 28782 ax-hfvadd 28783 ax-hv0cl 28786 ax-hfvmul 28788 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-map 8391 df-nn 11626 df-hlim 28755 df-sh 28990 df-ch 29004 df-span 29092 |
This theorem is referenced by: spancl 29119 spanss2 29128 spanid 29130 spanss 29131 shsval3i 29171 elspani 29326 |
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