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Mirrors > Home > HSE Home > Th. List > spanid | Structured version Visualization version GIF version |
Description: A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanid | ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shss 29572 | . . 3 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
2 | spanval 29695 | . . 3 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥}) |
4 | intmin 4899 | . 2 ⊢ (𝐴 ∈ Sℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | eqtrd 2778 | 1 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 ∩ 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: spanssoc 29711 shs0i 29811 spansn0 29903 span0 29904 spanuni 29906 spansnpji 29940 spanunsni 29941 spansnji 30008 shatomistici 30723 |
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