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Mirrors > Home > MPE Home > Th. List > nmof | Structured version Visualization version GIF version |
Description: The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
Ref | Expression |
---|---|
nmof | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmofval.1 | . . 3 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | eqid 2798 | . . 3 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2798 | . . 3 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
4 | eqid 2798 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
5 | 1, 2, 3, 4 | nmofval 23320 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))}, ℝ*, < ))) |
6 | ssrab2 4007 | . . . 4 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))} ⊆ (0[,)+∞) | |
7 | icossxr 12810 | . . . 4 ⊢ (0[,)+∞) ⊆ ℝ* | |
8 | 6, 7 | sstri 3924 | . . 3 ⊢ {𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))} ⊆ ℝ* |
9 | infxrcl 12714 | . . 3 ⊢ ({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))} ⊆ ℝ* → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))}, ℝ*, < ) ∈ ℝ*) | |
10 | 8, 9 | mp1i 13 | . 2 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ 𝑓 ∈ (𝑆 GrpHom 𝑇)) → inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑆)((norm‘𝑇)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑆)‘𝑥))}, ℝ*, < ) ∈ ℝ*) |
11 | 5, 10 | fmpt3d 6857 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 0cc0 10526 · cmul 10531 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 [,)cico 12728 Basecbs 16475 GrpHom cghm 18347 normcnm 23183 NrmGrpcngp 23184 normOp cnmo 23311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-ico 12732 df-nmo 23314 |
This theorem is referenced by: nmocl 23326 isnghm 23329 |
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