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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 6677 | . . . . 5 ⊢ (𝑥 = 0 → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘ 0 ))) | |
2 | fveq2 6672 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 )) | |
3 | 2 | oveq2d 7174 | . . . . 5 ⊢ (𝑥 = 0 → (𝐴 · (𝐿‘𝑥)) = (𝐴 · (𝐿‘ 0 ))) |
4 | 1, 3 | breq12d 5081 | . . . 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 470 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
7 | 0le0 11741 | . . . . . . 7 ⊢ 0 ≤ 0 | |
8 | nmolb2d.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | 8 | recnd 10671 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
10 | 9 | mul01d 10841 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 0) = 0) |
11 | 7, 10 | breqtrrid 5106 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴 · 0)) |
12 | nmolb2d.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
13 | nmolb2d.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑆) | |
14 | eqid 2823 | . . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
15 | 13, 14 | ghmid 18366 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = (0g‘𝑇)) |
16 | 12, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑇)) |
17 | 16 | fveq2d 6676 | . . . . . . 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 23240 | . . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
21 | 18, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
22 | 17, 21 | eqtrd 2858 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = 0) |
23 | nmolb2d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ NrmGrp) | |
24 | nmofval.3 | . . . . . . . . 9 ⊢ 𝐿 = (norm‘𝑆) | |
25 | 24, 13 | nm0 23240 | . . . . . . . 8 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
26 | 23, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘ 0 ) = 0) |
27 | 26 | oveq2d 7174 | . . . . . 6 ⊢ (𝜑 → (𝐴 · (𝐿‘ 0 )) = (𝐴 · 0)) |
28 | 11, 22, 27 | 3brtr4d 5100 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
29 | 28 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
30 | 4, 6, 29 | pm2.61ne 3104 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
31 | 30 | ralrimiva 3184 | . 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 23328 | . . 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 1380 | . 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 · cmul 10544 ≤ cle 10678 Basecbs 16485 0gc0g 16715 GrpHom cghm 18357 normcnm 23188 NrmGrpcngp 23189 normOp cnmo 23316 |
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-iun 4923 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-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-ico 12747 df-0g 16717 df-topgen 16719 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-ghm 18358 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-xms 22932 df-ms 22933 df-nm 23194 df-ngp 23195 df-nmo 23319 |
This theorem is referenced by: nmo0 23346 nmoco 23348 nmotri 23350 nmoid 23353 nmoleub2lem 23720 |
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