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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 24099 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ngpms 24100 | . . . . . . . . 9 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ MetSp) |
6 | msxms 23951 | . . . . . . . 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 23953 | . . . . . . 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 23936 | . . . . . 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 22792 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
17 | 14, 15, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
18 | nglmle.5 | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
19 | eqid 2732 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
20 | eqid 2732 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
21 | 18, 8, 19, 20, 9 | nmval2 24092 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
22 | 3, 17, 21 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
23 | 8, 19 | grpidcl 18846 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
25 | xmetsym 23844 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) | |
26 | 11, 17, 24, 25 | syl3anc 1371 | . . 3 ⊢ (𝜑 → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) |
27 | 22, 26 | eqtrd 2772 | . 2 ⊢ (𝜑 → (𝑁‘𝑃) = ((0g‘𝐺)𝐷𝑃)) |
28 | nnuz 12861 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
29 | 1zzd 12589 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
30 | nglmle.9 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐺 ∈ Grp) |
32 | nglmle.7 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
33 | 32 | ffvelcdmda 7083 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋) |
34 | 18, 8, 19, 20, 9 | nmval2 24092 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
35 | 31, 33, 34 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
36 | 11 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | 24 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0g‘𝐺) ∈ 𝑋) |
38 | xmetsym 23844 | . . . . . 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 1371 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
40 | 35, 39 | eqtrd 2772 | . . . 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 5171 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((0g‘𝐺)𝐷(𝐹‘𝑘)) ≤ 𝑅) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 24809 | . 2 ⊢ (𝜑 → ((0g‘𝐺)𝐷𝑃) ≤ 𝑅) |
44 | 27, 43 | eqbrtrd 5169 | 1 ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 × cxp 5673 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 1c1 11107 ℝ*cxr 11243 ≤ cle 11245 ℕcn 12208 Basecbs 17140 distcds 17202 0gc0g 17381 Grpcgrp 18815 ∞Metcxmet 20921 MetOpencmopn 20926 TopOnctopon 22403 ⇝𝑡clm 22721 ∞MetSpcxms 23814 MetSpcms 23815 normcnm 24076 NrmGrpcngp 24077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-psmet 20928 df-xmet 20929 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-lm 22724 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 |
This theorem is referenced by: (None) |
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