Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28981 | . . 3 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
2 | spanval 29104 | . . 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 4888 | . 2 ⊢ (𝐴 ∈ Sℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | eqtrd 2856 | 1 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {crab 3142 ⊆ wss 3935 ∩ cint 4868 ‘cfv 6349 ℋchba 28690 Sℋ csh 28699 spancspn 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-1cn 10589 ax-addcl 10591 ax-hilex 28770 ax-hfvadd 28771 ax-hv0cl 28774 ax-hfvmul 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-map 8402 df-nn 11633 df-hlim 28743 df-sh 28978 df-ch 28992 df-span 29080 |
This theorem is referenced by: spanssoc 29120 shs0i 29220 spansn0 29312 span0 29313 spanuni 29315 spansnpji 29349 spanunsni 29350 spansnji 29417 shatomistici 30132 |
Copyright terms: Public domain | W3C validator |