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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 22623 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ngpms 22624 | . . . . . . . . 9 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ MetSp) |
6 | msxms 22479 | . . . . . . . 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 22481 | . . . . . . 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 22464 | . . . . . 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 21322 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
17 | 14, 15, 16 | syl2anc 573 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
18 | nglmle.5 | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
19 | eqid 2771 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
20 | eqid 2771 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
21 | 18, 8, 19, 20, 9 | nmval2 22616 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
22 | 3, 17, 21 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
23 | 8, 19 | grpidcl 17658 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
25 | xmetsym 22372 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) | |
26 | 11, 17, 24, 25 | syl3anc 1476 | . . 3 ⊢ (𝜑 → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) |
27 | 22, 26 | eqtrd 2805 | . 2 ⊢ (𝜑 → (𝑁‘𝑃) = ((0g‘𝐺)𝐷𝑃)) |
28 | nnuz 11925 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
29 | 1zzd 11610 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
30 | nglmle.9 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 3 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐺 ∈ Grp) |
32 | nglmle.7 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
33 | 32 | ffvelrnda 6502 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋) |
34 | 18, 8, 19, 20, 9 | nmval2 22616 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
35 | 31, 33, 34 | syl2anc 573 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
36 | 11 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | 24 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0g‘𝐺) ∈ 𝑋) |
38 | xmetsym 22372 | . . . . . 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 1476 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
40 | 35, 39 | eqtrd 2805 | . . . 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 4810 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((0g‘𝐺)𝐷(𝐹‘𝑘)) ≤ 𝑅) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 23318 | . 2 ⊢ (𝜑 → ((0g‘𝐺)𝐷𝑃) ≤ 𝑅) |
44 | 27, 43 | eqbrtrd 4808 | 1 ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 × cxp 5247 ↾ cres 5251 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 1c1 10139 ℝ*cxr 10275 ≤ cle 10277 ℕcn 11222 Basecbs 16064 distcds 16158 0gc0g 16308 Grpcgrp 17630 ∞Metcxmt 19946 MetOpencmopn 19951 TopOnctopon 20935 ⇝𝑡clm 21251 ∞MetSpcxme 22342 MetSpcmt 22343 normcnm 22601 NrmGrpcngp 22602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-map 8011 df-pm 8012 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-inf 8505 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-n0 11495 df-z 11580 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-0g 16310 df-topgen 16312 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-psmet 19953 df-xmet 19954 df-bl 19956 df-mopn 19957 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-lm 21254 df-xms 22345 df-ms 22346 df-nm 22607 df-ngp 22608 |
This theorem is referenced by: (None) |
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