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Mirrors > Home > HSE Home > Th. List > hilcompl | Structured version Visualization version GIF version |
Description: Lemma used for derivation of the completeness axiom ax-hcompl 28981 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 28778; the 6th would be satisfied by eqid 2823; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 28694. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hilcompl.1 | ⊢ ℋ = (BaseSet‘𝑈) |
hilcompl.2 | ⊢ +ℎ = ( +𝑣 ‘𝑈) |
hilcompl.3 | ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
hilcompl.4 | ⊢ ·ih = (·𝑖OLD‘𝑈) |
hilcompl.5 | ⊢ 𝐷 = (IndMet‘𝑈) |
hilcompl.6 | ⊢ 𝐽 = (MetOpen‘𝐷) |
hilcompl.7 | ⊢ 𝑈 ∈ CHilOLD |
hilcompl.8 | ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) |
Ref | Expression |
---|---|
hilcompl | ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hilcompl.1 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) | |
2 | hilcompl.2 | . . 3 ⊢ +ℎ = ( +𝑣 ‘𝑈) | |
3 | hilcompl.3 | . . 3 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | |
4 | hilcompl.4 | . . 3 ⊢ ·ih = (·𝑖OLD‘𝑈) | |
5 | hilcompl.7 | . . . 4 ⊢ 𝑈 ∈ CHilOLD | |
6 | 5 | hlnvi 28671 | . . 3 ⊢ 𝑈 ∈ NrmCVec |
7 | 1, 2, 3, 4, 6 | hilhhi 28943 | . 2 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
8 | hilcompl.5 | . 2 ⊢ 𝐷 = (IndMet‘𝑈) | |
9 | hilcompl.6 | . 2 ⊢ 𝐽 = (MetOpen‘𝐷) | |
10 | hilcompl.8 | . 2 ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) | |
11 | 7, 8, 9, 10 | hhcmpl 28979 | 1 ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 MetOpencmopn 20537 ⇝𝑡clm 21836 Cauccau 23858 +𝑣 cpv 28364 BaseSetcba 28365 ·𝑠OLD cns 28366 IndMetcims 28370 ·𝑖OLDcdip 28479 CHilOLDchlo 28664 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 ·ih csp 28701 Cauchyccauold 28705 ⇝𝑣 chli 28706 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 ax-hilex 28778 ax-hfvadd 28779 ax-hvcom 28780 ax-hvass 28781 ax-hv0cl 28782 ax-hvaddid 28783 ax-hfvmul 28784 ax-hvmulid 28785 ax-hvmulass 28786 ax-hvdistr1 28787 ax-hvdistr2 28788 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 ax-his4 28864 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-lm 21839 df-cau 23861 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-cbn 28642 df-hlo 28665 df-hnorm 28747 df-hvsub 28750 df-hlim 28751 df-hcau 28752 |
This theorem is referenced by: (None) |
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