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Mirrors > Home > HSE Home > Th. List > spansnpji | Structured version Visualization version GIF version |
Description: A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spansnpj.1 | β’ π΄ β β |
spansnpj.2 | β’ π΅ β β |
Ref | Expression |
---|---|
spansnpji | β’ π΄ β (β₯β(spanβ{((projββ(β₯βπ΄))βπ΅)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansnpj.1 | . . 3 β’ π΄ β β | |
2 | ococss 30810 | . . 3 β’ (π΄ β β β π΄ β (β₯β(β₯βπ΄))) | |
3 | 1, 2 | ax-mp 5 | . 2 β’ π΄ β (β₯β(β₯βπ΄)) |
4 | occl 30821 | . . . . . . 7 β’ (π΄ β β β (β₯βπ΄) β Cβ ) | |
5 | 1, 4 | ax-mp 5 | . . . . . 6 β’ (β₯βπ΄) β Cβ |
6 | 5 | chssii 30748 | . . . . 5 β’ (β₯βπ΄) β β |
7 | spansnpj.2 | . . . . . . 7 β’ π΅ β β | |
8 | 5, 7 | pjclii 30938 | . . . . . 6 β’ ((projββ(β₯βπ΄))βπ΅) β (β₯βπ΄) |
9 | snssi 4812 | . . . . . 6 β’ (((projββ(β₯βπ΄))βπ΅) β (β₯βπ΄) β {((projββ(β₯βπ΄))βπ΅)} β (β₯βπ΄)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 β’ {((projββ(β₯βπ΄))βπ΅)} β (β₯βπ΄) |
11 | spanss 30865 | . . . . 5 β’ (((β₯βπ΄) β β β§ {((projββ(β₯βπ΄))βπ΅)} β (β₯βπ΄)) β (spanβ{((projββ(β₯βπ΄))βπ΅)}) β (spanβ(β₯βπ΄))) | |
12 | 6, 10, 11 | mp2an 689 | . . . 4 β’ (spanβ{((projββ(β₯βπ΄))βπ΅)}) β (spanβ(β₯βπ΄)) |
13 | 5 | chshii 30744 | . . . . 5 β’ (β₯βπ΄) β Sβ |
14 | spanid 30864 | . . . . 5 β’ ((β₯βπ΄) β Sβ β (spanβ(β₯βπ΄)) = (β₯βπ΄)) | |
15 | 13, 14 | ax-mp 5 | . . . 4 β’ (spanβ(β₯βπ΄)) = (β₯βπ΄) |
16 | 12, 15 | sseqtri 4019 | . . 3 β’ (spanβ{((projββ(β₯βπ΄))βπ΅)}) β (β₯βπ΄) |
17 | 5, 7 | pjhclii 30939 | . . . . 5 β’ ((projββ(β₯βπ΄))βπ΅) β β |
18 | 17 | spansnchi 31079 | . . . 4 β’ (spanβ{((projββ(β₯βπ΄))βπ΅)}) β Cβ |
19 | 18, 5 | chsscon3i 30978 | . . 3 β’ ((spanβ{((projββ(β₯βπ΄))βπ΅)}) β (β₯βπ΄) β (β₯β(β₯βπ΄)) β (β₯β(spanβ{((projββ(β₯βπ΄))βπ΅)}))) |
20 | 16, 19 | mpbi 229 | . 2 β’ (β₯β(β₯βπ΄)) β (β₯β(spanβ{((projββ(β₯βπ΄))βπ΅)})) |
21 | 3, 20 | sstri 3992 | 1 β’ π΄ β (β₯β(spanβ{((projββ(β₯βπ΄))βπ΅)})) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 β wcel 2105 β wss 3949 {csn 4629 βcfv 6544 βchba 30436 Sβ csh 30445 Cβ cch 30446 β₯cort 30447 spancspn 30449 projβcpjh 30454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-cc 10433 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 ax-hilex 30516 ax-hfvadd 30517 ax-hvcom 30518 ax-hvass 30519 ax-hv0cl 30520 ax-hvaddid 30521 ax-hfvmul 30522 ax-hvmulid 30523 ax-hvmulass 30524 ax-hvdistr1 30525 ax-hvdistr2 30526 ax-hvmul0 30527 ax-hfi 30596 ax-his1 30599 ax-his2 30600 ax-his3 30601 ax-his4 30602 ax-hcompl 30719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-acn 9940 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-cn 22952 df-cnp 22953 df-lm 22954 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cfil 25004 df-cau 25005 df-cmet 25006 df-grpo 30010 df-gid 30011 df-ginv 30012 df-gdiv 30013 df-ablo 30062 df-vc 30076 df-nv 30109 df-va 30112 df-ba 30113 df-sm 30114 df-0v 30115 df-vs 30116 df-nmcv 30117 df-ims 30118 df-dip 30218 df-ssp 30239 df-ph 30330 df-cbn 30380 df-hnorm 30485 df-hba 30486 df-hvsub 30488 df-hlim 30489 df-hcau 30490 df-sh 30724 df-ch 30738 df-oc 30769 df-ch0 30770 df-shs 30825 df-span 30826 df-pjh 30912 |
This theorem is referenced by: spansnji 31163 |
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