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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 22213 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
4 | ngpms 22214 | . . . . . . . . 9 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ MetSp) |
6 | msxms 22069 | . . . . . . . 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 22071 | . . . . . . 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 22054 | . . . . . 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 20911 | . . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹(⇝𝑡‘𝐽)𝑃) → 𝑃 ∈ 𝑋) | |
17 | 14, 15, 16 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
18 | nglmle.5 | . . . . 5 ⊢ 𝑁 = (norm‘𝐺) | |
19 | eqid 2610 | . . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
20 | eqid 2610 | . . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
21 | 18, 8, 19, 20, 9 | nmval2 22206 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑃 ∈ 𝑋) → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
22 | 3, 17, 21 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑁‘𝑃) = (𝑃𝐷(0g‘𝐺))) |
23 | 8, 19 | grpidcl 17273 | . . . . 5 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝑋) |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑋) |
25 | xmetsym 21962 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (0g‘𝐺) ∈ 𝑋) → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) | |
26 | 11, 17, 24, 25 | syl3anc 1318 | . . 3 ⊢ (𝜑 → (𝑃𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷𝑃)) |
27 | 22, 26 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝑁‘𝑃) = ((0g‘𝐺)𝐷𝑃)) |
28 | nnuz 11599 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
29 | 1zzd 11285 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
30 | nglmle.9 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 3 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐺 ∈ Grp) |
32 | nglmle.7 | . . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) | |
33 | 32 | ffvelrnda 6267 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ 𝑋) |
34 | 18, 8, 19, 20, 9 | nmval2 22206 | . . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝐹‘𝑘) ∈ 𝑋) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
35 | 31, 33, 34 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑁‘(𝐹‘𝑘)) = ((𝐹‘𝑘)𝐷(0g‘𝐺))) |
36 | 11 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
37 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (0g‘𝐺) ∈ 𝑋) |
38 | xmetsym 21962 | . . . . . 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 1318 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(0g‘𝐺)) = ((0g‘𝐺)𝐷(𝐹‘𝑘))) |
40 | 35, 39 | eqtrd 2644 | . . . 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 4607 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((0g‘𝐺)𝐷(𝐹‘𝑘)) ≤ 𝑅) |
43 | 28, 12, 11, 29, 15, 24, 30, 42 | lmle 22907 | . 2 ⊢ (𝜑 → ((0g‘𝐺)𝐷𝑃) ≤ 𝑅) |
44 | 27, 43 | eqbrtrd 4605 | 1 ⊢ (𝜑 → (𝑁‘𝑃) ≤ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1c1 9816 ℝ*cxr 9952 ≤ cle 9954 ℕcn 10897 Basecbs 15695 distcds 15777 0gc0g 15923 Grpcgrp 17245 ∞Metcxmt 19552 MetOpencmopn 19557 TopOnctopon 20518 ⇝𝑡clm 20840 ∞MetSpcxme 21932 MetSpcmt 21933 normcnm 22191 NrmGrpcngp 22192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-0g 15925 df-topgen 15927 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-lm 20843 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 |
This theorem is referenced by: (None) |
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