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Mirrors > Home > MPE Home > Th. List > nmoleub2b | Structured version Visualization version GIF version |
Description: The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.) |
Ref | Expression |
---|---|
nmoleub2.n | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmoleub2.v | ⊢ 𝑉 = (Base‘𝑆) |
nmoleub2.l | ⊢ 𝐿 = (norm‘𝑆) |
nmoleub2.m | ⊢ 𝑀 = (norm‘𝑇) |
nmoleub2.g | ⊢ 𝐺 = (Scalar‘𝑆) |
nmoleub2.w | ⊢ 𝐾 = (Base‘𝐺) |
nmoleub2.s | ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) |
nmoleub2.t | ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) |
nmoleub2.f | ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
nmoleub2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
nmoleub2.r | ⊢ (𝜑 → 𝑅 ∈ ℝ+) |
nmoleub2a.5 | ⊢ (𝜑 → ℚ ⊆ 𝐾) |
Ref | Expression |
---|---|
nmoleub2b | ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoleub2.n | . 2 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | nmoleub2.v | . 2 ⊢ 𝑉 = (Base‘𝑆) | |
3 | nmoleub2.l | . 2 ⊢ 𝐿 = (norm‘𝑆) | |
4 | nmoleub2.m | . 2 ⊢ 𝑀 = (norm‘𝑇) | |
5 | nmoleub2.g | . 2 ⊢ 𝐺 = (Scalar‘𝑆) | |
6 | nmoleub2.w | . 2 ⊢ 𝐾 = (Base‘𝐺) | |
7 | nmoleub2.s | . 2 ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) | |
8 | nmoleub2.t | . 2 ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) | |
9 | nmoleub2.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) | |
10 | nmoleub2.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
11 | nmoleub2.r | . 2 ⊢ (𝜑 → 𝑅 ∈ ℝ+) | |
12 | nmoleub2a.5 | . 2 ⊢ (𝜑 → ℚ ⊆ 𝐾) | |
13 | ltle 11309 | . 2 ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥) ≤ 𝑅)) | |
14 | idd 24 | . 2 ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥) < 𝑅)) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | nmoleub2lem2 24963 | 1 ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 ℝ*cxr 11254 < clt 11255 ≤ cle 11256 / cdiv 11878 ℚcq 12939 ℝ+crp 12981 Basecbs 17151 Scalarcsca 17207 LMHom clmhm 20863 normcnm 24405 NrmModcnlm 24409 normOp cnmo 24542 ℂModcclm 24909 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-fz 13492 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-topgen 17396 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-subg 19046 df-ghm 19135 df-cmn 19698 df-mgp 20036 df-ring 20136 df-cring 20137 df-subrg 20467 df-lmod 20704 df-lmhm 20866 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-cnfld 21234 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-xms 24146 df-ms 24147 df-nm 24411 df-ngp 24412 df-nlm 24415 df-nmo 24545 df-nghm 24546 df-clm 24910 |
This theorem is referenced by: nmhmcn 24967 |
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