Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nmolb2d | Structured version Visualization version GIF version |
Description: Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmofval.2 | ⊢ 𝑉 = (Base‘𝑆) |
nmofval.3 | ⊢ 𝐿 = (norm‘𝑆) |
nmofval.4 | ⊢ 𝑀 = (norm‘𝑇) |
nmolb2d.z | ⊢ 0 = (0g‘𝑆) |
nmolb2d.1 | ⊢ (𝜑 → 𝑆 ∈ NrmGrp) |
nmolb2d.2 | ⊢ (𝜑 → 𝑇 ∈ NrmGrp) |
nmolb2d.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
nmolb2d.4 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
nmolb2d.5 | ⊢ (𝜑 → 0 ≤ 𝐴) |
nmolb2d.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
Ref | Expression |
---|---|
nmolb2d | ⊢ (𝜑 → (𝑁‘𝐹) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘ 0 )) | |
2 | 1 | fveq2d 6195 | . . . . 5 ⊢ (𝑥 = 0 → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘ 0 ))) |
3 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 )) | |
4 | 3 | oveq2d 6666 | . . . . 5 ⊢ (𝑥 = 0 → (𝐴 · (𝐿‘𝑥)) = (𝐴 · (𝐿‘ 0 ))) |
5 | 2, 4 | breq12d 4666 | . . . 4 ⊢ (𝑥 = 0 → ((𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) ↔ (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 )))) |
6 | nmolb2d.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) | |
7 | 6 | anassrs 680 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
8 | 0le0 11110 | . . . . . . 7 ⊢ 0 ≤ 0 | |
9 | nmolb2d.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
10 | 9 | recnd 10068 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | 10 | mul01d 10235 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 0) = 0) |
12 | 8, 11 | syl5breqr 4691 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴 · 0)) |
13 | nmolb2d.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
14 | nmolb2d.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑆) | |
15 | eqid 2622 | . . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
16 | 14, 15 | ghmid 17666 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = (0g‘𝑇)) |
17 | 13, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑇)) |
18 | 17 | fveq2d 6195 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = (𝑀‘(0g‘𝑇))) |
19 | nmolb2d.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) | |
20 | nmofval.4 | . . . . . . . . 9 ⊢ 𝑀 = (norm‘𝑇) | |
21 | 20, 15 | nm0 22433 | . . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
22 | 19, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
23 | 18, 22 | eqtrd 2656 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = 0) |
24 | nmolb2d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ NrmGrp) | |
25 | nmofval.3 | . . . . . . . . 9 ⊢ 𝐿 = (norm‘𝑆) | |
26 | 25, 14 | nm0 22433 | . . . . . . . 8 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
27 | 24, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘ 0 ) = 0) |
28 | 27 | oveq2d 6666 | . . . . . 6 ⊢ (𝜑 → (𝐴 · (𝐿‘ 0 )) = (𝐴 · 0)) |
29 | 12, 23, 28 | 3brtr4d 4685 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
30 | 29 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
31 | 5, 7, 30 | pm2.61ne 2879 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
32 | 31 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
33 | nmolb2d.5 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
34 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
35 | nmofval.2 | . . . 4 ⊢ 𝑉 = (Base‘𝑆) | |
36 | 34, 35, 25, 20 | nmolb 22521 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
37 | 24, 19, 13, 9, 33, 36 | syl311anc 1340 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
38 | 32, 37 | mpd 15 | 1 ⊢ (𝜑 → (𝑁‘𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 Basecbs 15857 0gc0g 16100 GrpHom cghm 17657 normcnm 22381 NrmGrpcngp 22382 normOp cnmo 22509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ico 12181 df-0g 16102 df-topgen 16104 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ghm 17658 df-psmet 19738 df-xmet 19739 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-nmo 22512 |
This theorem is referenced by: nmo0 22539 nmoco 22541 nmotri 22543 nmoid 22546 nmoleub2lem 22914 |
Copyright terms: Public domain | W3C validator |