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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 28623 | . . 3 ⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ) | |
2 | spanval 28748 | . . 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 4718 | . 2 ⊢ (𝐴 ∈ Sℋ → ∩ {𝑥 ∈ Sℋ ∣ 𝐴 ⊆ 𝑥} = 𝐴) | |
5 | 3, 4 | eqtrd 2862 | 1 ⊢ (𝐴 ∈ Sℋ → (span‘𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 {crab 3122 ⊆ wss 3799 ∩ cint 4698 ‘cfv 6124 ℋchba 28332 Sℋ csh 28341 spancspn 28345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-1cn 10311 ax-addcl 10313 ax-hilex 28412 ax-hfvadd 28413 ax-hv0cl 28416 ax-hfvmul 28418 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-map 8125 df-nn 11352 df-hlim 28385 df-sh 28620 df-ch 28634 df-span 28724 |
This theorem is referenced by: spanssoc 28764 shs0i 28864 spansn0 28956 span0 28957 spanuni 28959 spansnpji 28993 spanunsni 28994 spansnji 29061 shatomistici 29776 |
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