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Mirrors > Home > MPE Home > Th. List > nmoge0 | Structured version Visualization version GIF version |
Description: The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
Ref | Expression |
---|---|
nmoge0 | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrege0 12569 | . . . . . 6 ⊢ (𝑟 ∈ (0[,)+∞) ↔ (𝑟 ∈ ℝ ∧ 0 ≤ 𝑟)) | |
2 | 1 | simprbi 492 | . . . . 5 ⊢ (𝑟 ∈ (0[,)+∞) → 0 ≤ 𝑟) |
3 | 2 | adantl 475 | . . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑟 ∈ (0[,)+∞)) → 0 ≤ 𝑟) |
4 | 3 | a1d 25 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑟 ∈ (0[,)+∞)) → (∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝐹‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥)) → 0 ≤ 𝑟)) |
5 | 4 | ralrimiva 3176 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝐹‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥)) → 0 ≤ 𝑟)) |
6 | 0xr 10404 | . . 3 ⊢ 0 ∈ ℝ* | |
7 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
8 | eqid 2826 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | eqid 2826 | . . . 4 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
10 | eqid 2826 | . . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
11 | 7, 8, 9, 10 | nmogelb 22891 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 0 ∈ ℝ*) → (0 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝐹‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥)) → 0 ≤ 𝑟))) |
12 | 6, 11 | mpan2 684 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (0 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝐹‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥)) → 0 ≤ 𝑟))) |
13 | 5, 12 | mpbird 249 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3118 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 ℝcr 10252 0cc0 10253 · cmul 10258 +∞cpnf 10389 ℝ*cxr 10391 ≤ cle 10393 [,)cico 12466 Basecbs 16223 GrpHom cghm 18009 normcnm 22752 NrmGrpcngp 22753 normOp cnmo 22880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-ico 12470 df-nmo 22883 |
This theorem is referenced by: isnghm3 22900 bddnghm 22901 nmoi 22903 nmoix 22904 nmo0 22910 nmoco 22912 nmotri 22914 nmoid 22917 nghmcn 22920 nmoleub2lem 23284 |
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