![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12568 | . . . . . 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 3175 | . 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 10403 | . . 3 ⊢ 0 ∈ ℝ* | |
7 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
8 | eqid 2825 | . . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | eqid 2825 | . . . 4 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
10 | eqid 2825 | . . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
11 | 7, 8, 9, 10 | nmogelb 22890 | . . 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 3117 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 ℝcr 10251 0cc0 10252 · cmul 10257 +∞cpnf 10388 ℝ*cxr 10390 ≤ cle 10392 [,)cico 12465 Basecbs 16222 GrpHom cghm 18008 normcnm 22751 NrmGrpcngp 22752 normOp cnmo 22879 |
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 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
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 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-ico 12469 df-nmo 22882 |
This theorem is referenced by: isnghm3 22899 bddnghm 22900 nmoi 22902 nmoix 22903 nmo0 22909 nmoco 22911 nmotri 22913 nmoid 22916 nghmcn 22919 nmoleub2lem 23283 |
Copyright terms: Public domain | W3C validator |