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Mirrors > Home > HSE Home > Th. List > spansni | Structured version Visualization version GIF version |
Description: The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spansn.1 | ⊢ 𝐴 ∈ ℋ |
Ref | Expression |
---|---|
spansni | ⊢ (span‘{𝐴}) = (⊥‘(⊥‘{𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansn.1 | . . 3 ⊢ 𝐴 ∈ ℋ | |
2 | snssi 4747 | . . 3 ⊢ (𝐴 ∈ ℋ → {𝐴} ⊆ ℋ) | |
3 | spanssoc 29699 | . . 3 ⊢ ({𝐴} ⊆ ℋ → (span‘{𝐴}) ⊆ (⊥‘(⊥‘{𝐴}))) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (span‘{𝐴}) ⊆ (⊥‘(⊥‘{𝐴})) |
5 | 1 | elexi 3450 | . . . . . . . . 9 ⊢ 𝐴 ∈ V |
6 | 5 | snss 4725 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑦 ↔ {𝐴} ⊆ 𝑦) |
7 | shmulcl 29568 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ Sℋ ∧ 𝑧 ∈ ℂ ∧ 𝐴 ∈ 𝑦) → (𝑧 ·ℎ 𝐴) ∈ 𝑦) | |
8 | 7 | 3expia 1120 | . . . . . . . . 9 ⊢ ((𝑦 ∈ Sℋ ∧ 𝑧 ∈ ℂ) → (𝐴 ∈ 𝑦 → (𝑧 ·ℎ 𝐴) ∈ 𝑦)) |
9 | 8 | ancoms 459 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 ∈ Sℋ ) → (𝐴 ∈ 𝑦 → (𝑧 ·ℎ 𝐴) ∈ 𝑦)) |
10 | 6, 9 | syl5bir 242 | . . . . . . 7 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 ∈ Sℋ ) → ({𝐴} ⊆ 𝑦 → (𝑧 ·ℎ 𝐴) ∈ 𝑦)) |
11 | eleq1 2828 | . . . . . . . 8 ⊢ (𝑥 = (𝑧 ·ℎ 𝐴) → (𝑥 ∈ 𝑦 ↔ (𝑧 ·ℎ 𝐴) ∈ 𝑦)) | |
12 | 11 | imbi2d 341 | . . . . . . 7 ⊢ (𝑥 = (𝑧 ·ℎ 𝐴) → (({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦) ↔ ({𝐴} ⊆ 𝑦 → (𝑧 ·ℎ 𝐴) ∈ 𝑦))) |
13 | 10, 12 | syl5ibrcom 246 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 ∈ Sℋ ) → (𝑥 = (𝑧 ·ℎ 𝐴) → ({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦))) |
14 | 13 | ralrimdva 3115 | . . . . 5 ⊢ (𝑧 ∈ ℂ → (𝑥 = (𝑧 ·ℎ 𝐴) → ∀𝑦 ∈ Sℋ ({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦))) |
15 | 14 | rexlimiv 3211 | . . . 4 ⊢ (∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴) → ∀𝑦 ∈ Sℋ ({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦)) |
16 | 1 | h1de2ci 29906 | . . . 4 ⊢ (𝑥 ∈ (⊥‘(⊥‘{𝐴})) ↔ ∃𝑧 ∈ ℂ 𝑥 = (𝑧 ·ℎ 𝐴)) |
17 | vex 3435 | . . . . . 6 ⊢ 𝑥 ∈ V | |
18 | 17 | elspani 29893 | . . . . 5 ⊢ ({𝐴} ⊆ ℋ → (𝑥 ∈ (span‘{𝐴}) ↔ ∀𝑦 ∈ Sℋ ({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦))) |
19 | 1, 2, 18 | mp2b 10 | . . . 4 ⊢ (𝑥 ∈ (span‘{𝐴}) ↔ ∀𝑦 ∈ Sℋ ({𝐴} ⊆ 𝑦 → 𝑥 ∈ 𝑦)) |
20 | 15, 16, 19 | 3imtr4i 292 | . . 3 ⊢ (𝑥 ∈ (⊥‘(⊥‘{𝐴})) → 𝑥 ∈ (span‘{𝐴})) |
21 | 20 | ssriv 3930 | . 2 ⊢ (⊥‘(⊥‘{𝐴})) ⊆ (span‘{𝐴}) |
22 | 4, 21 | eqssi 3942 | 1 ⊢ (span‘{𝐴}) = (⊥‘(⊥‘{𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 ⊆ wss 3892 {csn 4567 ‘cfv 6431 (class class class)co 7269 ℂcc 10862 ℋchba 29269 ·ℎ csm 29271 Sℋ csh 29278 ⊥cort 29280 spancspn 29282 |
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 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9369 ax-cc 10184 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 ax-pre-sup 10942 ax-addf 10943 ax-mulf 10944 ax-hilex 29349 ax-hfvadd 29350 ax-hvcom 29351 ax-hvass 29352 ax-hv0cl 29353 ax-hvaddid 29354 ax-hfvmul 29355 ax-hvmulid 29356 ax-hvmulass 29357 ax-hvdistr1 29358 ax-hvdistr2 29359 ax-hvmul0 29360 ax-hfi 29429 ax-his1 29432 ax-his2 29433 ax-his3 29434 ax-his4 29435 ax-hcompl 29552 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 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 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7702 df-1st 7818 df-2nd 7819 df-supp 7963 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-2o 8283 df-oadd 8286 df-omul 8287 df-er 8473 df-map 8592 df-pm 8593 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-fsupp 9099 df-fi 9140 df-sup 9171 df-inf 9172 df-oi 9239 df-card 9690 df-acn 9693 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-div 11625 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-q 12680 df-rp 12722 df-xneg 12839 df-xadd 12840 df-xmul 12841 df-ioo 13074 df-ico 13076 df-icc 13077 df-fz 13231 df-fzo 13374 df-fl 13502 df-seq 13712 df-exp 13773 df-hash 14035 df-cj 14800 df-re 14801 df-im 14802 df-sqrt 14936 df-abs 14937 df-clim 15187 df-rlim 15188 df-sum 15388 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-starv 16967 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-unif 16975 df-hom 16976 df-cco 16977 df-rest 17123 df-topn 17124 df-0g 17142 df-gsum 17143 df-topgen 17144 df-pt 17145 df-prds 17148 df-xrs 17203 df-qtop 17208 df-imas 17209 df-xps 17211 df-mre 17285 df-mrc 17286 df-acs 17288 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-submnd 18421 df-mulg 18691 df-cntz 18913 df-cmn 19378 df-psmet 20579 df-xmet 20580 df-met 20581 df-bl 20582 df-mopn 20583 df-fbas 20584 df-fg 20585 df-cnfld 20588 df-top 22033 df-topon 22050 df-topsp 22072 df-bases 22086 df-cld 22160 df-ntr 22161 df-cls 22162 df-nei 22239 df-cn 22368 df-cnp 22369 df-lm 22370 df-haus 22456 df-tx 22703 df-hmeo 22896 df-fil 22987 df-fm 23079 df-flim 23080 df-flf 23081 df-xms 23463 df-ms 23464 df-tms 23465 df-cfil 24409 df-cau 24410 df-cmet 24411 df-grpo 28843 df-gid 28844 df-ginv 28845 df-gdiv 28846 df-ablo 28895 df-vc 28909 df-nv 28942 df-va 28945 df-ba 28946 df-sm 28947 df-0v 28948 df-vs 28949 df-nmcv 28950 df-ims 28951 df-dip 29051 df-ssp 29072 df-ph 29163 df-cbn 29213 df-hnorm 29318 df-hba 29319 df-hvsub 29321 df-hlim 29322 df-hcau 29323 df-sh 29557 df-ch 29571 df-oc 29602 df-ch0 29603 df-span 29659 |
This theorem is referenced by: elspansni 29908 spansn 29909 |
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