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Mirrors > Home > MPE Home > Th. List > nmods | Structured version Visualization version GIF version |
Description: Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.) |
Ref | Expression |
---|---|
nmods.n | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
nmods.v | ⊢ 𝑉 = (Base‘𝑆) |
nmods.c | ⊢ 𝐶 = (dist‘𝑆) |
nmods.d | ⊢ 𝐷 = (dist‘𝑇) |
Ref | Expression |
---|---|
nmods | ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1167 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 NGHom 𝑇)) | |
2 | nghmrcl1 22868 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | |
3 | ngpgrp 22735 | . . . . 5 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ Grp) |
5 | nmods.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑆) | |
6 | eqid 2803 | . . . . 5 ⊢ (-g‘𝑆) = (-g‘𝑆) | |
7 | 5, 6 | grpsubcl 17815 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
8 | 4, 7 | syl3an1 1203 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
9 | nmods.n | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
10 | eqid 2803 | . . . 4 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
11 | eqid 2803 | . . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
12 | 9, 5, 10, 11 | nmoi 22864 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
13 | 1, 8, 12 | syl2anc 580 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) ≤ ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
14 | nghmrcl2 22869 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp) | |
15 | 14 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ NrmGrp) |
16 | nghmghm 22870 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
17 | 16 | 3ad2ant1 1164 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
18 | eqid 2803 | . . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
19 | 5, 18 | ghmf 17981 | . . . . . 6 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑉⟶(Base‘𝑇)) |
20 | 17, 19 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹:𝑉⟶(Base‘𝑇)) |
21 | simp2 1168 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
22 | 20, 21 | ffvelrnd 6590 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐴) ∈ (Base‘𝑇)) |
23 | simp3 1169 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
24 | 20, 23 | ffvelrnd 6590 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐵) ∈ (Base‘𝑇)) |
25 | eqid 2803 | . . . . 5 ⊢ (-g‘𝑇) = (-g‘𝑇) | |
26 | nmods.d | . . . . 5 ⊢ 𝐷 = (dist‘𝑇) | |
27 | 11, 18, 25, 26 | ngpds 22740 | . . . 4 ⊢ ((𝑇 ∈ NrmGrp ∧ (𝐹‘𝐴) ∈ (Base‘𝑇) ∧ (𝐹‘𝐵) ∈ (Base‘𝑇)) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
28 | 15, 22, 24, 27 | syl3anc 1491 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
29 | 5, 6, 25 | ghmsub 17985 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
30 | 16, 29 | syl3an1 1203 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
31 | 30 | fveq2d 6419 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵))) = ((norm‘𝑇)‘((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵)))) |
32 | 28, 31 | eqtr4d 2840 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) = ((norm‘𝑇)‘(𝐹‘(𝐴(-g‘𝑆)𝐵)))) |
33 | nmods.c | . . . . 5 ⊢ 𝐶 = (dist‘𝑆) | |
34 | 10, 5, 6, 33 | ngpds 22740 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
35 | 2, 34 | syl3an1 1203 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
36 | 35 | oveq2d 6898 | . 2 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐹) · (𝐴𝐶𝐵)) = ((𝑁‘𝐹) · ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵)))) |
37 | 13, 32, 36 | 3brtr4d 4879 | 1 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 class class class wbr 4847 ⟶wf 6101 ‘cfv 6105 (class class class)co 6882 · cmul 10233 ≤ cle 10368 Basecbs 16188 distcds 16280 Grpcgrp 17742 -gcsg 17744 GrpHom cghm 17974 normcnm 22713 NrmGrpcngp 22714 normOp cnmo 22841 NGHom cnghm 22842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2379 ax-ext 2781 ax-rep 4968 ax-sep 4979 ax-nul 4987 ax-pow 5039 ax-pr 5101 ax-un 7187 ax-cnex 10284 ax-resscn 10285 ax-1cn 10286 ax-icn 10287 ax-addcl 10288 ax-addrcl 10289 ax-mulcl 10290 ax-mulrcl 10291 ax-mulcom 10292 ax-addass 10293 ax-mulass 10294 ax-distr 10295 ax-i2m1 10296 ax-1ne0 10297 ax-1rid 10298 ax-rnegex 10299 ax-rrecex 10300 ax-cnre 10301 ax-pre-lttri 10302 ax-pre-lttrn 10303 ax-pre-ltadd 10304 ax-pre-mulgt0 10305 ax-pre-sup 10306 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2593 df-eu 2611 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ne 2976 df-nel 3079 df-ral 3098 df-rex 3099 df-reu 3100 df-rmo 3101 df-rab 3102 df-v 3391 df-sbc 3638 df-csb 3733 df-dif 3776 df-un 3778 df-in 3780 df-ss 3787 df-pss 3789 df-nul 4120 df-if 4282 df-pw 4355 df-sn 4373 df-pr 4375 df-tp 4377 df-op 4379 df-uni 4633 df-iun 4716 df-br 4848 df-opab 4910 df-mpt 4927 df-tr 4950 df-id 5224 df-eprel 5229 df-po 5237 df-so 5238 df-fr 5275 df-we 5277 df-xp 5322 df-rel 5323 df-cnv 5324 df-co 5325 df-dm 5326 df-rn 5327 df-res 5328 df-ima 5329 df-pred 5902 df-ord 5948 df-on 5949 df-lim 5950 df-suc 5951 df-iota 6068 df-fun 6107 df-fn 6108 df-f 6109 df-f1 6110 df-fo 6111 df-f1o 6112 df-fv 6113 df-riota 6843 df-ov 6885 df-oprab 6886 df-mpt2 6887 df-om 7304 df-1st 7405 df-2nd 7406 df-wrecs 7649 df-recs 7711 df-rdg 7749 df-er 7986 df-map 8101 df-en 8200 df-dom 8201 df-sdom 8202 df-sup 8594 df-inf 8595 df-pnf 10369 df-mnf 10370 df-xr 10371 df-ltxr 10372 df-le 10373 df-sub 10562 df-neg 10563 df-div 10981 df-nn 11317 df-2 11380 df-n0 11585 df-z 11671 df-uz 11935 df-q 12038 df-rp 12079 df-xneg 12197 df-xadd 12198 df-xmul 12199 df-ico 12434 df-0g 16421 df-topgen 16423 df-mgm 17561 df-sgrp 17603 df-mnd 17614 df-grp 17745 df-minusg 17746 df-sbg 17747 df-ghm 17975 df-psmet 20064 df-xmet 20065 df-met 20066 df-bl 20067 df-mopn 20068 df-top 21031 df-topon 21048 df-topsp 21070 df-bases 21083 df-xms 22457 df-ms 22458 df-nm 22719 df-ngp 22720 df-nmo 22844 df-nghm 22845 |
This theorem is referenced by: nghmcn 22881 |
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