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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 | β’ (π΄ β Sβ β (spanβπ΄) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shss 30958 | . . 3 β’ (π΄ β Sβ β π΄ β β) | |
2 | spanval 31081 | . . 3 β’ (π΄ β β β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) | |
3 | 1, 2 | syl 17 | . 2 β’ (π΄ β Sβ β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) |
4 | intmin 4963 | . 2 β’ (π΄ β Sβ β β© {π₯ β Sβ β£ π΄ β π₯} = π΄) | |
5 | 3, 4 | eqtrd 2764 | 1 β’ (π΄ β Sβ β (spanβπ΄) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {crab 3424 β wss 3941 β© cint 4941 βcfv 6534 βchba 30667 Sβ csh 30676 spancspn 30680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 ax-hilex 30747 ax-hfvadd 30748 ax-hv0cl 30751 ax-hfvmul 30753 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-map 8819 df-nn 12212 df-hlim 30720 df-sh 30955 df-ch 30969 df-span 31057 |
This theorem is referenced by: spanssoc 31097 shs0i 31197 spansn0 31289 span0 31290 spanuni 31292 spansnpji 31326 spanunsni 31327 spansnji 31394 shatomistici 32109 |
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