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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 23210 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ngpms 23211 | . . . . . . . . 9 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ MetSp) |
6 | msxms 23066 | . . . . . . . 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 23068 | . . . . . . 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 23051 | . . . . . 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 21907 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
17 | 14, 15, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
18 | nglmle.5 | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
19 | eqid 2823 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
20 | eqid 2823 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
21 | 18, 8, 19, 20, 9 | nmval2 23203 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
22 | 3, 17, 21 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
23 | 8, 19 | grpidcl 18133 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
25 | xmetsym 22959 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) | |
26 | 11, 17, 24, 25 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) |
27 | 22, 26 | eqtrd 2858 | . 2 ⊢ (𝜑 → (𝑁‘𝑃) = ((0g‘𝐺)𝐷𝑃)) |
28 | nnuz 12284 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
29 | 1zzd 12016 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
30 | nglmle.9 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐺 ∈ Grp) |
32 | nglmle.7 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
33 | 32 | ffvelrnda 6853 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋) |
34 | 18, 8, 19, 20, 9 | nmval2 23203 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
35 | 31, 33, 34 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
36 | 11 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | 24 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0g‘𝐺) ∈ 𝑋) |
38 | xmetsym 22959 | . . . . . 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 1367 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
40 | 35, 39 | eqtrd 2858 | . . . 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 5092 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((0g‘𝐺)𝐷(𝐹‘𝑘)) ≤ 𝑅) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 23906 | . 2 ⊢ (𝜑 → ((0g‘𝐺)𝐷𝑃) ≤ 𝑅) |
44 | 27, 43 | eqbrtrd 5090 | 1 ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 × cxp 5555 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1c1 10540 ℝ*cxr 10676 ≤ cle 10678 ℕcn 11640 Basecbs 16485 distcds 16576 0gc0g 16715 Grpcgrp 18105 ∞Metcxmet 20532 MetOpencmopn 20537 TopOnctopon 21520 ⇝𝑡clm 21836 ∞MetSpcxms 22929 MetSpcms 22930 normcnm 23188 NrmGrpcngp 23189 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-lm 21839 df-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 |
This theorem is referenced by: (None) |
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