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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 31033 | . . 3 β’ (π΄ β Sβ β π΄ β β) | |
2 | spanval 31156 | . . 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 4971 | . 2 β’ (π΄ β Sβ β β© {π₯ β Sβ β£ π΄ β π₯} = π΄) | |
5 | 3, 4 | eqtrd 2768 | 1 β’ (π΄ β Sβ β (spanβπ΄) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {crab 3429 β wss 3947 β© cint 4949 βcfv 6548 βchba 30742 Sβ csh 30751 spancspn 30755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-1cn 11197 ax-addcl 11199 ax-hilex 30822 ax-hfvadd 30823 ax-hv0cl 30826 ax-hfvmul 30828 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-map 8847 df-nn 12244 df-hlim 30795 df-sh 31030 df-ch 31044 df-span 31132 |
This theorem is referenced by: spanssoc 31172 shs0i 31272 spansn0 31364 span0 31365 spanuni 31367 spansnpji 31401 spanunsni 31402 spansnji 31469 shatomistici 32184 |
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