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Mirrors > Home > MPE Home > Th. List > nglmle | Structured version Visualization version GIF version |
Description: If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.) |
Ref | Expression |
---|---|
nglmle.1 | ⊢ 𝑋 = (Base‘𝐺) |
nglmle.2 | ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) |
nglmle.3 | ⊢ 𝐽 = (MetOpen‘𝐷) |
nglmle.5 | ⊢ 𝑁 = (norm‘𝐺) |
nglmle.6 | ⊢ (𝜑 → 𝐺 ∈ NrmGrp) |
nglmle.7 | ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) |
nglmle.8 | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) |
nglmle.9 | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
nglmle.10 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) ≤ 𝑅) |
Ref | Expression |
---|---|
nglmle | ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nglmle.6 | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ NrmGrp) | |
2 | ngpgrp 23497 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ngpms 23498 | . . . . . . . . 9 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ MetSp) |
6 | msxms 23352 | . . . . . . . 8 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ∞MetSp) |
8 | nglmle.1 | . . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺) | |
9 | nglmle.2 | . . . . . . . 8 ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋)) | |
10 | 8, 9 | xmsxmet 23354 | . . . . . . 7 ⊢ (𝐺 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋)) |
11 | 7, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
12 | nglmle.3 | . . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷) | |
13 | 12 | mopntopon 23337 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
14 | 11, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
15 | nglmle.8 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | |
16 | lmcl 22194 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
17 | 14, 15, 16 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
18 | nglmle.5 | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
19 | eqid 2737 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
20 | eqid 2737 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
21 | 18, 8, 19, 20, 9 | nmval2 23490 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
22 | 3, 17, 21 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
23 | 8, 19 | grpidcl 18395 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
25 | xmetsym 23245 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) | |
26 | 11, 17, 24, 25 | syl3anc 1373 | . . 3 ⊢ (𝜑 → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) |
27 | 22, 26 | eqtrd 2777 | . 2 ⊢ (𝜑 → (𝑁‘𝑃) = ((0g‘𝐺)𝐷𝑃)) |
28 | nnuz 12477 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
29 | 1zzd 12208 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
30 | nglmle.9 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐺 ∈ Grp) |
32 | nglmle.7 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
33 | 32 | ffvelrnda 6904 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋) |
34 | 18, 8, 19, 20, 9 | nmval2 23490 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
35 | 31, 33, 34 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
36 | 11 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | 24 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0g‘𝐺) ∈ 𝑋) |
38 | xmetsym 23245 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) | |
39 | 36, 33, 37, 38 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
40 | 35, 39 | eqtrd 2777 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
41 | nglmle.10 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) ≤ 𝑅) | |
42 | 40, 41 | eqbrtrrd 5077 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((0g‘𝐺)𝐷(𝐹‘𝑘)) ≤ 𝑅) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 24198 | . 2 ⊢ (𝜑 → ((0g‘𝐺)𝐷𝑃) ≤ 𝑅) |
44 | 27, 43 | eqbrtrd 5075 | 1 ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 class class class wbr 5053 × cxp 5549 ↾ cres 5553 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 1c1 10730 ℝ*cxr 10866 ≤ cle 10868 ℕcn 11830 Basecbs 16760 distcds 16811 0gc0g 16944 Grpcgrp 18365 ∞Metcxmet 20348 MetOpencmopn 20353 TopOnctopon 21807 ⇝𝑡clm 22123 ∞MetSpcxms 23215 MetSpcms 23216 normcnm 23474 NrmGrpcngp 23475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-0g 16946 df-topgen 16948 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-psmet 20355 df-xmet 20356 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-lm 22126 df-xms 23218 df-ms 23219 df-nm 23480 df-ngp 23481 |
This theorem is referenced by: (None) |
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