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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 | hlimadd.3 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) | |
2 | 1 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℋ) |
3 | hlimadd.4 | . . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) | |
4 | 3 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℋ) |
5 | hvaddcl 28789 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℋ ∧ (𝐺‘𝑛) ∈ ℋ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) | |
6 | 2, 4, 5 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) |
7 | hlimadd.7 | . . . 4 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) | |
8 | 6, 7 | fmptd 6878 | . . 3 ⊢ (𝜑 → 𝐻:ℕ⟶ ℋ) |
9 | ax-hilex 28776 | . . . 4 ⊢ ℋ ∈ V | |
10 | nnex 11644 | . . . 4 ⊢ ℕ ∈ V | |
11 | 9, 10 | elmap 8435 | . . 3 ⊢ (𝐻 ∈ ( ℋ ↑m ℕ) ↔ 𝐻:ℕ⟶ ℋ) |
12 | 8, 11 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐻 ∈ ( ℋ ↑m ℕ)) |
13 | nnuz 12282 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 12014 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | eqid 2821 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
16 | eqid 2821 | . . . . . 6 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
17 | 15, 16 | hhims 28949 | . . . . 5 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
18 | 15, 17 | hhxmet 28952 | . . . 4 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
19 | eqid 2821 | . . . . 5 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
20 | 19 | mopntopon 23049 | . . . 4 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
21 | 18, 20 | mp1i 13 | . . 3 ⊢ (𝜑 → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
22 | hlimadd.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) | |
23 | 15 | hhnv 28942 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
24 | df-hba 28746 | . . . . . . 7 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
25 | 15, 23, 24, 17, 19 | h2hlm 28757 | . . . . . 6 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
26 | resss 5878 | . . . . . 6 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
27 | 25, 26 | eqsstri 4001 | . . . . 5 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
28 | 27 | ssbri 5111 | . . . 4 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
29 | 22, 28 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
30 | hlimadd.6 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) | |
31 | 27 | ssbri 5111 | . . . 4 ⊢ (𝐺 ⇝𝑣 𝐵 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
32 | 30, 31 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
33 | 15 | hhva 28943 | . . . . 5 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
34 | 17, 19, 33 | vacn 28471 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
35 | 23, 34 | mp1i 13 | . . 3 ⊢ (𝜑 → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
36 | 13, 14, 21, 21, 1, 3, 29, 32, 35, 7 | lmcn2 22257 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵)) |
37 | 25 | breqi 5072 | . . 3 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ 𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵)) |
38 | ovex 7189 | . . . 4 ⊢ (𝐴 +ℎ 𝐵) ∈ V | |
39 | 38 | brresi 5862 | . . 3 ⊢ (𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
40 | 37, 39 | bitri 277 | . 2 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
41 | 12, 36, 40 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 〈cop 4573 class class class wbr 5066 ↦ cmpt 5146 ↾ cres 5557 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 1c1 10538 ℕcn 11638 ∞Metcxmet 20530 MetOpencmopn 20535 TopOnctopon 21518 Cn ccn 21832 ⇝𝑡clm 21834 ×t ctx 22168 NrmCVeccnv 28361 ℋchba 28696 +ℎ cva 28697 ·ℎ csm 28698 normℎcno 28700 −ℎ cmv 28702 ⇝𝑣 chli 28704 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvmulass 28784 ax-hvdistr1 28785 ax-hvdistr2 28786 ax-hvmul0 28787 ax-hfi 28856 ax-his1 28859 ax-his2 28860 ax-his3 28861 ax-his4 28862 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cn 21835 df-cnp 21836 df-lm 21837 df-tx 22170 df-hmeo 22363 df-xms 22930 df-tms 22932 df-grpo 28270 df-gid 28271 df-ginv 28272 df-gdiv 28273 df-ablo 28322 df-vc 28336 df-nv 28369 df-va 28372 df-ba 28373 df-sm 28374 df-0v 28375 df-vs 28376 df-nmcv 28377 df-ims 28378 df-hnorm 28745 df-hba 28746 df-hvsub 28748 df-hlim 28749 |
This theorem is referenced by: chscllem4 29417 |
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