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Mirrors > Home > HSE Home > Th. List > spanval | Structured version Visualization version GIF version |
Description: Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spanval | β’ (π΄ β β β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-span 31139 | . 2 β’ span = (π¦ β π« β β¦ β© {π₯ β Sβ β£ π¦ β π₯}) | |
2 | sseq1 4007 | . . . 4 β’ (π¦ = π΄ β (π¦ β π₯ β π΄ β π₯)) | |
3 | 2 | rabbidv 3438 | . . 3 β’ (π¦ = π΄ β {π₯ β Sβ β£ π¦ β π₯} = {π₯ β Sβ β£ π΄ β π₯}) |
4 | 3 | inteqd 4958 | . 2 β’ (π¦ = π΄ β β© {π₯ β Sβ β£ π¦ β π₯} = β© {π₯ β Sβ β£ π΄ β π₯}) |
5 | ax-hilex 30829 | . . . 4 β’ β β V | |
6 | 5 | elpw2 5351 | . . 3 β’ (π΄ β π« β β π΄ β β) |
7 | 6 | biimpri 227 | . 2 β’ (π΄ β β β π΄ β π« β) |
8 | helsh 31075 | . . . 4 β’ β β Sβ | |
9 | sseq2 4008 | . . . . 5 β’ (π₯ = β β (π΄ β π₯ β π΄ β β)) | |
10 | 9 | rspcev 3611 | . . . 4 β’ (( β β Sβ β§ π΄ β β) β βπ₯ β Sβ π΄ β π₯) |
11 | 8, 10 | mpan 688 | . . 3 β’ (π΄ β β β βπ₯ β Sβ π΄ β π₯) |
12 | intexrab 5346 | . . 3 β’ (βπ₯ β Sβ π΄ β π₯ β β© {π₯ β Sβ β£ π΄ β π₯} β V) | |
13 | 11, 12 | sylib 217 | . 2 β’ (π΄ β β β β© {π₯ β Sβ β£ π΄ β π₯} β V) |
14 | 1, 4, 7, 13 | fvmptd3 7033 | 1 β’ (π΄ β β β (spanβπ΄) = β© {π₯ β Sβ β£ π΄ β π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwrex 3067 {crab 3430 Vcvv 3473 β wss 3949 π« cpw 4606 β© cint 4953 βcfv 6553 βchba 30749 Sβ csh 30758 spancspn 30762 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-1cn 11204 ax-addcl 11206 ax-hilex 30829 ax-hfvadd 30830 ax-hv0cl 30833 ax-hfvmul 30835 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-map 8853 df-nn 12251 df-hlim 30802 df-sh 31037 df-ch 31051 df-span 31139 |
This theorem is referenced by: spancl 31166 spanss2 31175 spanid 31177 spanss 31178 shsval3i 31218 elspani 31373 |
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