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Mirrors > Home > HSE Home > Th. List > hlimadd | Structured version Visualization version GIF version |
Description: Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimadd.3 | ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) |
hlimadd.4 | ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) |
hlimadd.5 | ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) |
hlimadd.6 | ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) |
hlimadd.7 | ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) |
Ref | Expression |
---|---|
hlimadd | ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11723 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 11408 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | eqid 2622 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
4 | eqid 2622 | . . . . . 6 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
5 | 3, 4 | hhims 28029 | . . . . 5 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
6 | 3, 5 | hhxmet 28032 | . . . 4 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
7 | eqid 2622 | . . . . 5 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
8 | 7 | mopntopon 22244 | . . . 4 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
9 | 6, 8 | mp1i 13 | . . 3 ⊢ (𝜑 → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
10 | hlimadd.3 | . . 3 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) | |
11 | hlimadd.4 | . . 3 ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) | |
12 | hlimadd.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) | |
13 | 3 | hhnv 28022 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
14 | df-hba 27826 | . . . . . . 7 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
15 | 3, 13, 14, 5, 7 | h2hlm 27837 | . . . . . 6 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑𝑚 ℕ)) |
16 | resss 5422 | . . . . . 6 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑𝑚 ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
17 | 15, 16 | eqsstri 3635 | . . . . 5 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
18 | 17 | ssbri 4697 | . . . 4 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
19 | 12, 18 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
20 | hlimadd.6 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) | |
21 | 17 | ssbri 4697 | . . . 4 ⊢ (𝐺 ⇝𝑣 𝐵 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
22 | 20, 21 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
23 | 3 | hhva 28023 | . . . . 5 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
24 | 5, 7, 23 | vacn 27549 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
25 | 13, 24 | mp1i 13 | . . 3 ⊢ (𝜑 → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
26 | hlimadd.7 | . . 3 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) | |
27 | 1, 2, 9, 9, 10, 11, 19, 22, 25, 26 | lmcn2 21452 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵)) |
28 | 10 | ffvelrnda 6359 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℋ) |
29 | 11 | ffvelrnda 6359 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℋ) |
30 | hvaddcl 27869 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℋ ∧ (𝐺‘𝑛) ∈ ℋ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) | |
31 | 28, 29, 30 | syl2anc 693 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) |
32 | 31, 26 | fmptd 6385 | . . 3 ⊢ (𝜑 → 𝐻:ℕ⟶ ℋ) |
33 | ax-hilex 27856 | . . . 4 ⊢ ℋ ∈ V | |
34 | nnex 11026 | . . . 4 ⊢ ℕ ∈ V | |
35 | 33, 34 | elmap 7886 | . . 3 ⊢ (𝐻 ∈ ( ℋ ↑𝑚 ℕ) ↔ 𝐻:ℕ⟶ ℋ) |
36 | 32, 35 | sylibr 224 | . 2 ⊢ (𝜑 → 𝐻 ∈ ( ℋ ↑𝑚 ℕ)) |
37 | 15 | breqi 4659 | . . 3 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ 𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑𝑚 ℕ))(𝐴 +ℎ 𝐵)) |
38 | ovex 6678 | . . . 4 ⊢ (𝐴 +ℎ 𝐵) ∈ V | |
39 | 38 | brres 5402 | . . 3 ⊢ (𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑𝑚 ℕ))(𝐴 +ℎ 𝐵) ↔ (𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵) ∧ 𝐻 ∈ ( ℋ ↑𝑚 ℕ))) |
40 | 37, 39 | bitri 264 | . 2 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ (𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵) ∧ 𝐻 ∈ ( ℋ ↑𝑚 ℕ))) |
41 | 27, 36, 40 | sylanbrc 698 | 1 ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 ↾ cres 5116 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 1c1 9937 ℕcn 11020 ∞Metcxmt 19731 MetOpencmopn 19736 TopOnctopon 20715 Cn ccn 21028 ⇝𝑡clm 21030 ×t ctx 21363 NrmCVeccnv 27439 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 normℎcno 27780 −ℎ cmv 27782 ⇝𝑣 chli 27784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-lm 21033 df-tx 21365 df-hmeo 21558 df-xms 22125 df-tms 22127 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 |
This theorem is referenced by: chscllem4 28499 |
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