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Mirrors > Home > MPE Home > Th. List > nmo0 | Structured version Visualization version GIF version |
Description: The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.) |
Ref | Expression |
---|---|
nmo0.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmo0.2 | ⊢ 𝑉 = (Base‘𝑆) |
nmo0.3 | ⊢ 0 = (0g‘𝑇) |
Ref | Expression |
---|---|
nmo0 | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmo0.1 | . . 3 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | nmo0.2 | . . 3 ⊢ 𝑉 = (Base‘𝑆) | |
3 | eqid 2760 | . . 3 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
4 | eqid 2760 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
5 | eqid 2760 | . . 3 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
6 | simpl 474 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑆 ∈ NrmGrp) | |
7 | simpr 479 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑇 ∈ NrmGrp) | |
8 | ngpgrp 22604 | . . . 4 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
9 | ngpgrp 22604 | . . . 4 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ Grp) | |
10 | nmo0.3 | . . . . 5 ⊢ 0 = (0g‘𝑇) | |
11 | 10, 2 | 0ghm 17875 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
12 | 8, 9, 11 | syl2an 495 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) |
13 | 0red 10233 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ∈ ℝ) | |
14 | 0le0 11302 | . . . 4 ⊢ 0 ≤ 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ≤ 0) |
16 | fvex 6362 | . . . . . . . . 9 ⊢ (0g‘𝑇) ∈ V | |
17 | 10, 16 | eqeltri 2835 | . . . . . . . 8 ⊢ 0 ∈ V |
18 | 17 | fvconst2 6633 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ((𝑉 × { 0 })‘𝑥) = 0 ) |
19 | 18 | ad2antrl 766 | . . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((𝑉 × { 0 })‘𝑥) = 0 ) |
20 | 19 | fveq2d 6356 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) = ((norm‘𝑇)‘ 0 )) |
21 | 4, 10 | nm0 22634 | . . . . . 6 ⊢ (𝑇 ∈ NrmGrp → ((norm‘𝑇)‘ 0 ) = 0) |
22 | 21 | ad2antlr 765 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘ 0 ) = 0) |
23 | 20, 22 | eqtrd 2794 | . . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) = 0) |
24 | 2, 3 | nmcl 22621 | . . . . . . . 8 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ((norm‘𝑆)‘𝑥) ∈ ℝ) |
25 | 24 | ad2ant2r 800 | . . . . . . 7 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℝ) |
26 | 25 | recnd 10260 | . . . . . 6 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑆)‘𝑥) ∈ ℂ) |
27 | 26 | mul02d 10426 | . . . . 5 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → (0 · ((norm‘𝑆)‘𝑥)) = 0) |
28 | 14, 27 | syl5breqr 4842 | . . . 4 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → 0 ≤ (0 · ((norm‘𝑆)‘𝑥))) |
29 | 23, 28 | eqbrtrd 4826 | . . 3 ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ (0g‘𝑆))) → ((norm‘𝑇)‘((𝑉 × { 0 })‘𝑥)) ≤ (0 · ((norm‘𝑆)‘𝑥))) |
30 | 1, 2, 3, 4, 5, 6, 7, 12, 13, 15, 29 | nmolb2d 22723 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) ≤ 0) |
31 | 1 | nmoge0 22726 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘(𝑉 × { 0 }))) |
32 | 12, 31 | mpd3an3 1574 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 0 ≤ (𝑁‘(𝑉 × { 0 }))) |
33 | 1 | nmocl 22725 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ (𝑉 × { 0 }) ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘(𝑉 × { 0 })) ∈ ℝ*) |
34 | 12, 33 | mpd3an3 1574 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) ∈ ℝ*) |
35 | 0xr 10278 | . . 3 ⊢ 0 ∈ ℝ* | |
36 | xrletri3 12178 | . . 3 ⊢ (((𝑁‘(𝑉 × { 0 })) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝑁‘(𝑉 × { 0 })) = 0 ↔ ((𝑁‘(𝑉 × { 0 })) ≤ 0 ∧ 0 ≤ (𝑁‘(𝑉 × { 0 }))))) | |
37 | 34, 35, 36 | sylancl 697 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ((𝑁‘(𝑉 × { 0 })) = 0 ↔ ((𝑁‘(𝑉 × { 0 })) ≤ 0 ∧ 0 ≤ (𝑁‘(𝑉 × { 0 }))))) |
38 | 30, 32, 37 | mpbir2and 995 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 Vcvv 3340 {csn 4321 class class class wbr 4804 × cxp 5264 ‘cfv 6049 (class class class)co 6813 ℝcr 10127 0cc0 10128 · cmul 10133 ℝ*cxr 10265 ≤ cle 10267 Basecbs 16059 0gc0g 16302 Grpcgrp 17623 GrpHom cghm 17858 normcnm 22582 NrmGrpcngp 22583 normOp cnmo 22710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ico 12374 df-0g 16304 df-topgen 16306 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-grp 17626 df-ghm 17859 df-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-xms 22326 df-ms 22327 df-nm 22588 df-ngp 22589 df-nmo 22713 |
This theorem is referenced by: nmoeq0 22741 0nghm 22746 idnghm 22748 |
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