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Mirrors > Home > HSE Home > Th. List > spansnmul | Structured version Visualization version GIF version |
Description: A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spansnmul | β’ ((π΄ β β β§ π΅ β β) β (π΅ Β·β π΄) β (spanβ{π΄})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spansnsh 30302 | . . . 4 β’ (π΄ β β β (spanβ{π΄}) β Sβ ) | |
2 | spansnid 30304 | . . . 4 β’ (π΄ β β β π΄ β (spanβ{π΄})) | |
3 | 1, 2 | jca 513 | . . 3 β’ (π΄ β β β ((spanβ{π΄}) β Sβ β§ π΄ β (spanβ{π΄}))) |
4 | shmulcl 29959 | . . . . 5 β’ (((spanβ{π΄}) β Sβ β§ π΅ β β β§ π΄ β (spanβ{π΄})) β (π΅ Β·β π΄) β (spanβ{π΄})) | |
5 | 4 | 3com12 1124 | . . . 4 β’ ((π΅ β β β§ (spanβ{π΄}) β Sβ β§ π΄ β (spanβ{π΄})) β (π΅ Β·β π΄) β (spanβ{π΄})) |
6 | 5 | 3expb 1121 | . . 3 β’ ((π΅ β β β§ ((spanβ{π΄}) β Sβ β§ π΄ β (spanβ{π΄}))) β (π΅ Β·β π΄) β (spanβ{π΄})) |
7 | 3, 6 | sylan2 594 | . 2 β’ ((π΅ β β β§ π΄ β β) β (π΅ Β·β π΄) β (spanβ{π΄})) |
8 | 7 | ancoms 460 | 1 β’ ((π΄ β β β§ π΅ β β) β (π΅ Β·β π΄) β (spanβ{π΄})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 {csn 4585 βcfv 6492 (class class class)co 7350 βcc 10983 βchba 29660 Β·β csm 29662 Sβ csh 29669 spancspn 29673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cc 10305 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 ax-hilex 29740 ax-hfvadd 29741 ax-hvcom 29742 ax-hvass 29743 ax-hv0cl 29744 ax-hvaddid 29745 ax-hfvmul 29746 ax-hvmulid 29747 ax-hvmulass 29748 ax-hvdistr1 29749 ax-hvdistr2 29750 ax-hvmul0 29751 ax-hfi 29820 ax-his1 29823 ax-his2 29824 ax-his3 29825 ax-his4 29826 ax-hcompl 29943 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-acn 9812 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-q 12803 df-rp 12845 df-xneg 12962 df-xadd 12963 df-xmul 12964 df-ioo 13197 df-ico 13199 df-icc 13200 df-fz 13354 df-fzo 13497 df-fl 13626 df-seq 13836 df-exp 13897 df-hash 14159 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-clim 15305 df-rlim 15306 df-sum 15506 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-starv 17083 df-sca 17084 df-vsca 17085 df-ip 17086 df-tset 17087 df-ple 17088 df-ds 17090 df-unif 17091 df-hom 17092 df-cco 17093 df-rest 17239 df-topn 17240 df-0g 17258 df-gsum 17259 df-topgen 17260 df-pt 17261 df-prds 17264 df-xrs 17319 df-qtop 17324 df-imas 17325 df-xps 17327 df-mre 17401 df-mrc 17402 df-acs 17404 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-submnd 18537 df-mulg 18807 df-cntz 19030 df-cmn 19494 df-psmet 20712 df-xmet 20713 df-met 20714 df-bl 20715 df-mopn 20716 df-fbas 20717 df-fg 20718 df-cnfld 20721 df-top 22166 df-topon 22183 df-topsp 22205 df-bases 22219 df-cld 22293 df-ntr 22294 df-cls 22295 df-nei 22372 df-cn 22501 df-cnp 22502 df-lm 22503 df-haus 22589 df-tx 22836 df-hmeo 23029 df-fil 23120 df-fm 23212 df-flim 23213 df-flf 23214 df-xms 23596 df-ms 23597 df-tms 23598 df-cfil 24542 df-cau 24543 df-cmet 24544 df-grpo 29234 df-gid 29235 df-ginv 29236 df-gdiv 29237 df-ablo 29286 df-vc 29300 df-nv 29333 df-va 29336 df-ba 29337 df-sm 29338 df-0v 29339 df-vs 29340 df-nmcv 29341 df-ims 29342 df-dip 29442 df-ssp 29463 df-ph 29554 df-cbn 29604 df-hnorm 29709 df-hba 29710 df-hvsub 29712 df-hlim 29713 df-hcau 29714 df-sh 29948 df-ch 29962 df-oc 29993 df-ch0 29994 df-span 30050 |
This theorem is referenced by: spanunsni 30320 |
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