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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 7447 | . . . . 5 ⊢ (𝑥 = 0 → (𝐴 · (𝐿‘𝑥)) = (𝐴 · (𝐿‘ 0 ))) |
| 4 | 1, 3 | breq12d 5156 | . . . 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 12367 | . . . . . . 7 ⊢ 0 ≤ 0 | |
| 8 | nmolb2d.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 9 | 8 | recnd 11289 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 10 | 9 | mul01d 11460 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 0) = 0) |
| 11 | 7, 10 | breqtrrid 5181 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴 · 0)) |
| 12 | nmolb2d.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
| 13 | nmolb2d.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑆) | |
| 14 | eqid 2737 | . . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
| 15 | 13, 14 | ghmid 19240 | . . . . . . . . 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 24642 | . . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
| 21 | 18, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
| 22 | 17, 21 | eqtrd 2777 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = 0) |
| 23 | nmolb2d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ NrmGrp) | |
| 24 | nmofval.3 | . . . . . . . . 9 ⊢ 𝐿 = (norm‘𝑆) | |
| 25 | 24, 13 | nm0 24642 | . . . . . . . 8 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
| 26 | 23, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘ 0 ) = 0) |
| 27 | 26 | oveq2d 7447 | . . . . . 6 ⊢ (𝜑 → (𝐴 · (𝐿‘ 0 )) = (𝐴 · 0)) |
| 28 | 11, 22, 27 | 3brtr4d 5175 | . . . . 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 3027 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
| 31 | 30 | ralrimiva 3146 | . 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 24738 | . . 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 1386 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 · cmul 11160 ≤ cle 11296 Basecbs 17247 0gc0g 17484 GrpHom cghm 19230 normcnm 24589 NrmGrpcngp 24590 normOp cnmo 24726 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ghm 19231 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nmo 24729 |
| This theorem is referenced by: nmo0 24756 nmoco 24758 nmotri 24760 nmoid 24763 nmoleub2lem 25147 |
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