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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 1132 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 NGHom 𝑇)) | |
2 | nghmrcl1 23341 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp) | |
3 | ngpgrp 23208 | . . . . 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 2821 | . . . . 5 ⊢ (-g‘𝑆) = (-g‘𝑆) | |
7 | 5, 6 | grpsubcl 18179 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
8 | 4, 7 | syl3an1 1159 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(-g‘𝑆)𝐵) ∈ 𝑉) |
9 | nmods.n | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
10 | eqid 2821 | . . . 4 ⊢ (norm‘𝑆) = (norm‘𝑆) | |
11 | eqid 2821 | . . . 4 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
12 | 9, 5, 10, 11 | nmoi 23337 | . . 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 586 | . 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 23342 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp) | |
15 | 14 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ NrmGrp) |
16 | nghmghm 23343 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
17 | 16 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
18 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝑇) = (Base‘𝑇) | |
19 | 5, 18 | ghmf 18362 | . . . . . 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 1133 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
22 | 20, 21 | ffvelrnd 6852 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐴) ∈ (Base‘𝑇)) |
23 | simp3 1134 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) | |
24 | 20, 23 | ffvelrnd 6852 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘𝐵) ∈ (Base‘𝑇)) |
25 | eqid 2821 | . . . . 5 ⊢ (-g‘𝑇) = (-g‘𝑇) | |
26 | nmods.d | . . . . 5 ⊢ 𝐷 = (dist‘𝑇) | |
27 | 11, 18, 25, 26 | ngpds 23213 | . . . 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 1367 | . . 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 18366 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
30 | 16, 29 | syl3an1 1159 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐹‘(𝐴(-g‘𝑆)𝐵)) = ((𝐹‘𝐴)(-g‘𝑇)(𝐹‘𝐵))) |
31 | 30 | fveq2d 6674 | . . 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 2859 | . 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 23213 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
35 | 2, 34 | syl3an1 1159 | . . 3 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴𝐶𝐵) = ((norm‘𝑆)‘(𝐴(-g‘𝑆)𝐵))) |
36 | 35 | oveq2d 7172 | . 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 5098 | 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 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 · cmul 10542 ≤ cle 10676 Basecbs 16483 distcds 16574 Grpcgrp 18103 -gcsg 18105 GrpHom cghm 18355 normcnm 23186 NrmGrpcngp 23187 normOp cnmo 23314 NGHom cnghm 23315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ico 12745 df-0g 16715 df-topgen 16717 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-ghm 18356 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-nmo 23317 df-nghm 23318 |
This theorem is referenced by: nghmcn 23354 |
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