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Mirrors > Home > MPE Home > Th. List > hlcompl | Structured version Visualization version GIF version |
Description: Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlcompl.1 | β’ π· = (IndMetβπ) |
hlcompl.2 | β’ π½ = (MetOpenβπ·) |
Ref | Expression |
---|---|
hlcompl | β’ ((π β CHilOLD β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (BaseSetβπ) = (BaseSetβπ) | |
2 | hlcompl.1 | . . 3 β’ π· = (IndMetβπ) | |
3 | 1, 2 | hlcmet 29878 | . 2 β’ (π β CHilOLD β π· β (CMetβ(BaseSetβπ))) |
4 | hlcompl.2 | . . 3 β’ π½ = (MetOpenβπ·) | |
5 | 4 | cmetcau 24669 | . 2 β’ ((π· β (CMetβ(BaseSetβπ)) β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) |
6 | 3, 5 | sylan 581 | 1 β’ ((π β CHilOLD β§ πΉ β (Cauβπ·)) β πΉ β dom (βπ‘βπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 dom cdm 5634 βcfv 6497 MetOpencmopn 20802 βπ‘clm 22593 Cauccau 24633 CMetccmet 24634 BaseSetcba 29570 IndMetcims 29575 CHilOLDchlo 29869 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13276 df-rest 17309 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-bases 22312 df-ntr 22387 df-nei 22465 df-lm 22596 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-cfil 24635 df-cau 24636 df-cmet 24637 df-cbn 29847 df-hlo 29870 |
This theorem is referenced by: axhcompl-zf 29982 |
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