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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 29671 | . 2 ⊢ span = (𝑦 ∈ 𝒫 ℋ ↦ ∩ {𝑥 ∈ Sℋ ∣ 𝑦 ⊆ 𝑥}) | |
2 | sseq1 3946 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥)) | |
3 | 2 | rabbidv 3414 | . . 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 29361 | . . . 4 ⊢ ℋ ∈ V | |
6 | 5 | elpw2 5269 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ) |
7 | 6 | biimpri 227 | . 2 ⊢ (𝐴 ⊆ ℋ → 𝐴 ∈ 𝒫 ℋ) |
8 | helsh 29607 | . . . 4 ⊢ ℋ ∈ Sℋ | |
9 | sseq2 3947 | . . . . 5 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ)) | |
10 | 9 | rspcev 3561 | . . . 4 ⊢ (( ℋ ∈ Sℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
11 | 8, 10 | mpan 687 | . . 3 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Sℋ 𝐴 ⊆ 𝑥) |
12 | intexrab 5264 | . . 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 6898 | 1 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 {crab 3068 Vcvv 3432 ⊆ wss 3887 𝒫 cpw 4533 ∩ cint 4879 ‘cfv 6433 ℋchba 29281 Sℋ csh 29290 spancspn 29294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-1cn 10929 ax-addcl 10931 ax-hilex 29361 ax-hfvadd 29362 ax-hv0cl 29365 ax-hfvmul 29367 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-map 8617 df-nn 11974 df-hlim 29334 df-sh 29569 df-ch 29583 df-span 29671 |
This theorem is referenced by: spancl 29698 spanss2 29707 spanid 29709 spanss 29710 shsval3i 29750 elspani 29905 |
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