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Mirrors > Home > MPE Home > Th. List > nmogelb | Structured version Visualization version GIF version |
Description: Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmofval.2 | ⊢ 𝑉 = (Base‘𝑆) |
nmofval.3 | ⊢ 𝐿 = (norm‘𝑆) |
nmofval.4 | ⊢ 𝑀 = (norm‘𝑇) |
Ref | Expression |
---|---|
nmogelb | ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | nmofval.2 | . . . 4 ⊢ 𝑉 = (Base‘𝑆) | |
3 | nmofval.3 | . . . 4 ⊢ 𝐿 = (norm‘𝑆) | |
4 | nmofval.4 | . . . 4 ⊢ 𝑀 = (norm‘𝑇) | |
5 | 1, 2, 3, 4 | nmoval 24751 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )) |
6 | 5 | breq2d 5159 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐴 ≤ (𝑁‘𝐹) ↔ 𝐴 ≤ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ))) |
7 | ssrab2 4089 | . . . . 5 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ (0[,)+∞) | |
8 | icossxr 13468 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ* | |
9 | 7, 8 | sstri 4004 | . . . 4 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ* |
10 | infxrgelb 13373 | . . . 4 ⊢ (({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))} ⊆ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ↔ ∀𝑠 ∈ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}𝐴 ≤ 𝑠)) | |
11 | 9, 10 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ↔ ∀𝑠 ∈ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}𝐴 ≤ 𝑠)) |
12 | breq2 5151 | . . . 4 ⊢ (𝑠 = 𝑟 → (𝐴 ≤ 𝑠 ↔ 𝐴 ≤ 𝑟)) | |
13 | 12 | ralrab2 3706 | . . 3 ⊢ (∀𝑠 ∈ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}𝐴 ≤ 𝑠 ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟)) |
14 | 11, 13 | bitrdi 287 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ≤ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟))) |
15 | 6, 14 | sylan9bb 509 | 1 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 {crab 3432 ⊆ wss 3962 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 infcinf 9478 0cc0 11152 · cmul 11157 +∞cpnf 11289 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 [,)cico 13385 Basecbs 17244 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-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-id 5582 df-po 5596 df-so 5597 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-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-1st 8012 df-2nd 8013 df-er 8743 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-ico 13389 df-nmo 24744 |
This theorem is referenced by: nmolb 24753 nmoge0 24757 nmoi 24764 |
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