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Mirrors > Home > HSE Home > Th. List > hhcau | Structured version Visualization version GIF version |
Description: The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhlm.1 | β’ π = β¨β¨ +β , Β·β β©, normββ© |
hhlm.2 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
hhcau | β’ Cauchy = ((Cauβπ·) β© ( β βm β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhlm.1 | . 2 β’ π = β¨β¨ +β , Β·β β©, normββ© | |
2 | 1 | hhnv 30962 | . 2 β’ π β NrmCVec |
3 | 1 | hhba 30964 | . 2 β’ β = (BaseSetβπ) |
4 | hhlm.2 | . 2 β’ π· = (IndMetβπ) | |
5 | 1, 2, 3, 4 | h2hcau 30776 | 1 β’ Cauchy = ((Cauβπ·) β© ( β βm β)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β© cin 3943 β¨cop 4630 βcfv 6542 (class class class)co 7414 βm cmap 8836 βcn 12234 Cauccau 25168 IndMetcims 30388 βchba 30716 +β cva 30717 Β·β csm 30718 normβcno 30720 Cauchyccauold 30723 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210 ax-hilex 30796 ax-hfvadd 30797 ax-hvcom 30798 ax-hvass 30799 ax-hv0cl 30800 ax-hvaddid 30801 ax-hfvmul 30802 ax-hvmulid 30803 ax-hvmulass 30804 ax-hvdistr1 30805 ax-hvdistr2 30806 ax-hvmul0 30807 ax-hfi 30876 ax-his1 30879 ax-his2 30880 ax-his3 30881 ax-his4 30882 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-xneg 13116 df-xadd 13117 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-cau 25171 df-grpo 30290 df-gid 30291 df-ginv 30292 df-gdiv 30293 df-ablo 30342 df-vc 30356 df-nv 30389 df-va 30392 df-ba 30393 df-sm 30394 df-0v 30395 df-vs 30396 df-nmcv 30397 df-ims 30398 df-hnorm 30765 df-hvsub 30768 df-hcau 30770 |
This theorem is referenced by: hhcmpl 30997 hhcms 31000 hlimcaui 31033 hhsscms 31075 |
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