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Mirrors > Home > HSE Home > Th. List > spanssoc | Structured version Visualization version GIF version |
Description: The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanssoc | ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (⊥‘(⊥‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocss 30227 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | |
2 | ocss 30227 | . . . 4 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ℋ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ℋ) |
4 | ococss 30235 | . . 3 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | |
5 | spanss 30290 | . . 3 ⊢ (((⊥‘(⊥‘𝐴)) ⊆ ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴))) → (span‘𝐴) ⊆ (span‘(⊥‘(⊥‘𝐴)))) | |
6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (span‘(⊥‘(⊥‘𝐴)))) |
7 | ocsh 30225 | . . 3 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ Sℋ ) | |
8 | spanid 30289 | . . 3 ⊢ ((⊥‘(⊥‘𝐴)) ∈ Sℋ → (span‘(⊥‘(⊥‘𝐴))) = (⊥‘(⊥‘𝐴))) | |
9 | 1, 7, 8 | 3syl 18 | . 2 ⊢ (𝐴 ⊆ ℋ → (span‘(⊥‘(⊥‘𝐴))) = (⊥‘(⊥‘𝐴))) |
10 | 6, 9 | sseqtrd 3984 | 1 ⊢ (𝐴 ⊆ ℋ → (span‘𝐴) ⊆ (⊥‘(⊥‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ‘cfv 6496 ℋchba 29861 Sℋ csh 29870 ⊥cort 29872 spancspn 29874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-hilex 29941 ax-hfvadd 29942 ax-hv0cl 29945 ax-hfvmul 29947 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-cj 14984 df-re 14985 df-im 14986 df-hlim 29914 df-sh 30149 df-ch 30163 df-oc 30194 df-span 30251 |
This theorem is referenced by: spansni 30499 |
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