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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 | 2fveq3 6911 | . . . . 5 ⊢ (𝑥 = 0 → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘ 0 ))) | |
2 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 )) | |
3 | 2 | oveq2d 7446 | . . . . 5 ⊢ (𝑥 = 0 → (𝐴 · (𝐿‘𝑥)) = (𝐴 · (𝐿‘ 0 ))) |
4 | 1, 3 | breq12d 5160 | . . . 4 ⊢ (𝑥 = 0 → ((𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) ↔ (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 )))) |
5 | nmolb2d.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) | |
6 | 5 | anassrs 467 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
7 | 0le0 12364 | . . . . . . 7 ⊢ 0 ≤ 0 | |
8 | nmolb2d.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8 | recnd 11286 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | 9 | mul01d 11457 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 0) = 0) |
11 | 7, 10 | breqtrrid 5185 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴 · 0)) |
12 | nmolb2d.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
13 | nmolb2d.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑆) | |
14 | eqid 2734 | . . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
15 | 13, 14 | ghmid 19252 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = (0g‘𝑇)) |
16 | 12, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑇)) |
17 | 16 | fveq2d 6910 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = (𝑀‘(0g‘𝑇))) |
18 | nmolb2d.2 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ NrmGrp) | |
19 | nmofval.4 | . . . . . . . . 9 ⊢ 𝑀 = (norm‘𝑇) | |
20 | 19, 14 | nm0 24657 | . . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
21 | 18, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
22 | 17, 21 | eqtrd 2774 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = 0) |
23 | nmolb2d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ NrmGrp) | |
24 | nmofval.3 | . . . . . . . . 9 ⊢ 𝐿 = (norm‘𝑆) | |
25 | 24, 13 | nm0 24657 | . . . . . . . 8 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
26 | 23, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘ 0 ) = 0) |
27 | 26 | oveq2d 7446 | . . . . . 6 ⊢ (𝜑 → (𝐴 · (𝐿‘ 0 )) = (𝐴 · 0)) |
28 | 11, 22, 27 | 3brtr4d 5179 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
29 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
30 | 4, 6, 29 | pm2.61ne 3024 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
31 | 30 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
32 | nmolb2d.5 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
33 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
34 | nmofval.2 | . . . 4 ⊢ 𝑉 = (Base‘𝑆) | |
35 | 33, 34, 24, 19 | nmolb 24753 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
36 | 23, 18, 12, 8, 32, 35 | syl311anc 1383 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴)) |
37 | 31, 36 | mpd 15 | 1 ⊢ (𝜑 → (𝑁‘𝐹) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 · cmul 11157 ≤ cle 11293 Basecbs 17244 0gc0g 17485 GrpHom cghm 19242 normcnm 24604 NrmGrpcngp 24605 normOp cnmo 24741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ico 13389 df-0g 17487 df-topgen 17489 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-ghm 19243 df-psmet 21373 df-xmet 21374 df-bl 21376 df-mopn 21377 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-xms 24345 df-ms 24346 df-nm 24610 df-ngp 24611 df-nmo 24744 |
This theorem is referenced by: nmo0 24771 nmoco 24773 nmotri 24775 nmoid 24778 nmoleub2lem 25160 |
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