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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 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐹‘𝑥) = (𝐹‘ 0 )) | |
2 | 1 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 0 → (𝑀‘(𝐹‘𝑥)) = (𝑀‘(𝐹‘ 0 ))) |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑥 = 0 → (𝐿‘𝑥) = (𝐿‘ 0 )) | |
4 | 3 | oveq2d 6565 | . . . . 5 ⊢ (𝑥 = 0 → (𝐴 · (𝐿‘𝑥)) = (𝐴 · (𝐿‘ 0 ))) |
5 | 2, 4 | breq12d 4596 | . . . 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 678 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ 𝑥 ≠ 0 ) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
8 | 0le0 10987 | . . . . . . 7 ⊢ 0 ≤ 0 | |
9 | nmolb2d.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
10 | 9 | recnd 9947 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
11 | 10 | mul01d 10114 | . . . . . . 7 ⊢ (𝜑 → (𝐴 · 0) = 0) |
12 | 8, 11 | syl5breqr 4621 | . . . . . 6 ⊢ (𝜑 → 0 ≤ (𝐴 · 0)) |
13 | nmolb2d.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
14 | nmolb2d.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑆) | |
15 | eqid 2610 | . . . . . . . . . 10 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
16 | 14, 15 | ghmid 17489 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = (0g‘𝑇)) |
17 | 13, 16 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝑇)) |
18 | 17 | fveq2d 6107 | . . . . . . 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 22243 | . . . . . . . 8 ⊢ (𝑇 ∈ NrmGrp → (𝑀‘(0g‘𝑇)) = 0) |
22 | 19, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘(0g‘𝑇)) = 0) |
23 | 18, 22 | eqtrd 2644 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) = 0) |
24 | nmolb2d.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ NrmGrp) | |
25 | nmofval.3 | . . . . . . . . 9 ⊢ 𝐿 = (norm‘𝑆) | |
26 | 25, 14 | nm0 22243 | . . . . . . . 8 ⊢ (𝑆 ∈ NrmGrp → (𝐿‘ 0 ) = 0) |
27 | 24, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿‘ 0 ) = 0) |
28 | 27 | oveq2d 6565 | . . . . . 6 ⊢ (𝜑 → (𝐴 · (𝐿‘ 0 )) = (𝐴 · 0)) |
29 | 12, 23, 28 | 3brtr4d 4615 | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
30 | 29 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘ 0 )) ≤ (𝐴 · (𝐿‘ 0 ))) |
31 | 5, 7, 30 | pm2.61ne 2867 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥))) |
32 | 31 | ralrimiva 2949 | . 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 22331 | . . 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 1332 | . 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 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 · cmul 9820 ≤ cle 9954 Basecbs 15695 0gc0g 15923 GrpHom cghm 17480 normcnm 22191 NrmGrpcngp 22192 normOp cnmo 22319 |
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-iun 4457 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-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-ico 12052 df-0g 15925 df-topgen 15927 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ghm 17481 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-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-nmo 22322 |
This theorem is referenced by: nmo0 22349 nmoco 22351 nmotri 22353 nmoid 22356 nmoleub2lem 22722 |
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