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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 29148 | . . 3 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
2 | spanval 29271 | . . 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 4857 | . 2 ⊢ (𝐴 ∈ Sℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | eqtrd 2774 | 1 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {crab 3058 ⊆ wss 3844 ∩ cint 4837 ‘cfv 6340 ℋchba 28857 Sℋ csh 28866 spancspn 28870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-1cn 10676 ax-addcl 10678 ax-hilex 28937 ax-hfvadd 28938 ax-hv0cl 28941 ax-hfvmul 28943 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-map 8442 df-nn 11720 df-hlim 28910 df-sh 29145 df-ch 29159 df-span 29247 |
This theorem is referenced by: spanssoc 29287 shs0i 29387 spansn0 29479 span0 29480 spanuni 29482 spansnpji 29516 spanunsni 29517 spansnji 29584 shatomistici 30299 |
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