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Mirrors > Home > HSE Home > Th. List > hhmet | Structured version Visualization version GIF version |
Description: The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhnv.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhims2.2 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
hhmet | ⊢ 𝐷 ∈ (Met‘ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hhnv.1 | . . 3 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | 1 | hhnv 29048 | . 2 ⊢ 𝑈 ∈ NrmCVec |
3 | 1 | hhba 29050 | . . 3 ⊢ ℋ = (BaseSet‘𝑈) |
4 | hhims2.2 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
5 | 3, 4 | imsmet 28574 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘ ℋ)) |
6 | 2, 5 | ax-mp 5 | 1 ⊢ 𝐷 ∈ (Met‘ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2112 〈cop 4529 ‘cfv 6336 Metcmet 20153 NrmCVeccnv 28467 IndMetcims 28474 ℋchba 28802 +ℎ cva 28803 ·ℎ csm 28804 normℎcno 28806 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-cnex 10632 ax-resscn 10633 ax-1cn 10634 ax-icn 10635 ax-addcl 10636 ax-addrcl 10637 ax-mulcl 10638 ax-mulrcl 10639 ax-mulcom 10640 ax-addass 10641 ax-mulass 10642 ax-distr 10643 ax-i2m1 10644 ax-1ne0 10645 ax-1rid 10646 ax-rnegex 10647 ax-rrecex 10648 ax-cnre 10649 ax-pre-lttri 10650 ax-pre-lttrn 10651 ax-pre-ltadd 10652 ax-pre-mulgt0 10653 ax-pre-sup 10654 ax-addf 10655 ax-mulf 10656 ax-hilex 28882 ax-hfvadd 28883 ax-hvcom 28884 ax-hvass 28885 ax-hv0cl 28886 ax-hvaddid 28887 ax-hfvmul 28888 ax-hvmulid 28889 ax-hvmulass 28890 ax-hvdistr1 28891 ax-hvdistr2 28892 ax-hvmul0 28893 ax-hfi 28962 ax-his1 28965 ax-his2 28966 ax-his3 28967 ax-his4 28968 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-iun 4886 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-om 7581 df-1st 7694 df-2nd 7695 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-er 8300 df-map 8419 df-en 8529 df-dom 8530 df-sdom 8531 df-sup 8940 df-pnf 10716 df-mnf 10717 df-xr 10718 df-ltxr 10719 df-le 10720 df-sub 10911 df-neg 10912 df-div 11337 df-nn 11676 df-2 11738 df-3 11739 df-4 11740 df-n0 11936 df-z 12022 df-uz 12284 df-rp 12432 df-seq 13420 df-exp 13481 df-cj 14507 df-re 14508 df-im 14509 df-sqrt 14643 df-abs 14644 df-met 20161 df-grpo 28376 df-gid 28377 df-ginv 28378 df-gdiv 28379 df-ablo 28428 df-vc 28442 df-nv 28475 df-va 28478 df-ba 28479 df-sm 28480 df-0v 28481 df-vs 28482 df-nmcv 28483 df-ims 28484 df-hnorm 28851 df-hvsub 28854 |
This theorem is referenced by: hhxmet 29058 hilmet 29077 hhcms 29086 |
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