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Mirrors > Home > MPE Home > Th. List > nmolb2d | Structured version Visualization version GIF version |
Description: Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | β’ π = (π normOp π) |
nmofval.2 | β’ π = (Baseβπ) |
nmofval.3 | β’ πΏ = (normβπ) |
nmofval.4 | β’ π = (normβπ) |
nmolb2d.z | β’ 0 = (0gβπ) |
nmolb2d.1 | β’ (π β π β NrmGrp) |
nmolb2d.2 | β’ (π β π β NrmGrp) |
nmolb2d.3 | β’ (π β πΉ β (π GrpHom π)) |
nmolb2d.4 | β’ (π β π΄ β β) |
nmolb2d.5 | β’ (π β 0 β€ π΄) |
nmolb2d.6 | β’ ((π β§ (π₯ β π β§ π₯ β 0 )) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
Ref | Expression |
---|---|
nmolb2d | β’ (π β (πβπΉ) β€ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6907 | . . . . 5 β’ (π₯ = 0 β (πβ(πΉβπ₯)) = (πβ(πΉβ 0 ))) | |
2 | fveq2 6902 | . . . . . 6 β’ (π₯ = 0 β (πΏβπ₯) = (πΏβ 0 )) | |
3 | 2 | oveq2d 7442 | . . . . 5 β’ (π₯ = 0 β (π΄ Β· (πΏβπ₯)) = (π΄ Β· (πΏβ 0 ))) |
4 | 1, 3 | breq12d 5165 | . . . 4 β’ (π₯ = 0 β ((πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯)) β (πβ(πΉβ 0 )) β€ (π΄ Β· (πΏβ 0 )))) |
5 | nmolb2d.6 | . . . . 5 β’ ((π β§ (π₯ β π β§ π₯ β 0 )) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) | |
6 | 5 | anassrs 466 | . . . 4 β’ (((π β§ π₯ β π) β§ π₯ β 0 ) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
7 | 0le0 12351 | . . . . . . 7 β’ 0 β€ 0 | |
8 | nmolb2d.4 | . . . . . . . . 9 β’ (π β π΄ β β) | |
9 | 8 | recnd 11280 | . . . . . . . 8 β’ (π β π΄ β β) |
10 | 9 | mul01d 11451 | . . . . . . 7 β’ (π β (π΄ Β· 0) = 0) |
11 | 7, 10 | breqtrrid 5190 | . . . . . 6 β’ (π β 0 β€ (π΄ Β· 0)) |
12 | nmolb2d.3 | . . . . . . . . 9 β’ (π β πΉ β (π GrpHom π)) | |
13 | nmolb2d.z | . . . . . . . . . 10 β’ 0 = (0gβπ) | |
14 | eqid 2728 | . . . . . . . . . 10 β’ (0gβπ) = (0gβπ) | |
15 | 13, 14 | ghmid 19183 | . . . . . . . . 9 β’ (πΉ β (π GrpHom π) β (πΉβ 0 ) = (0gβπ)) |
16 | 12, 15 | syl 17 | . . . . . . . 8 β’ (π β (πΉβ 0 ) = (0gβπ)) |
17 | 16 | fveq2d 6906 | . . . . . . 7 β’ (π β (πβ(πΉβ 0 )) = (πβ(0gβπ))) |
18 | nmolb2d.2 | . . . . . . . 8 β’ (π β π β NrmGrp) | |
19 | nmofval.4 | . . . . . . . . 9 β’ π = (normβπ) | |
20 | 19, 14 | nm0 24558 | . . . . . . . 8 β’ (π β NrmGrp β (πβ(0gβπ)) = 0) |
21 | 18, 20 | syl 17 | . . . . . . 7 β’ (π β (πβ(0gβπ)) = 0) |
22 | 17, 21 | eqtrd 2768 | . . . . . 6 β’ (π β (πβ(πΉβ 0 )) = 0) |
23 | nmolb2d.1 | . . . . . . . 8 β’ (π β π β NrmGrp) | |
24 | nmofval.3 | . . . . . . . . 9 β’ πΏ = (normβπ) | |
25 | 24, 13 | nm0 24558 | . . . . . . . 8 β’ (π β NrmGrp β (πΏβ 0 ) = 0) |
26 | 23, 25 | syl 17 | . . . . . . 7 β’ (π β (πΏβ 0 ) = 0) |
27 | 26 | oveq2d 7442 | . . . . . 6 β’ (π β (π΄ Β· (πΏβ 0 )) = (π΄ Β· 0)) |
28 | 11, 22, 27 | 3brtr4d 5184 | . . . . 5 β’ (π β (πβ(πΉβ 0 )) β€ (π΄ Β· (πΏβ 0 ))) |
29 | 28 | adantr 479 | . . . 4 β’ ((π β§ π₯ β π) β (πβ(πΉβ 0 )) β€ (π΄ Β· (πΏβ 0 ))) |
30 | 4, 6, 29 | pm2.61ne 3024 | . . 3 β’ ((π β§ π₯ β π) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
31 | 30 | ralrimiva 3143 | . 2 β’ (π β βπ₯ β π (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
32 | nmolb2d.5 | . . 3 β’ (π β 0 β€ π΄) | |
33 | nmofval.1 | . . . 4 β’ π = (π normOp π) | |
34 | nmofval.2 | . . . 4 β’ π = (Baseβπ) | |
35 | 33, 34, 24, 19 | nmolb 24654 | . . 3 β’ (((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β§ π΄ β β β§ 0 β€ π΄) β (βπ₯ β π (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯)) β (πβπΉ) β€ π΄)) |
36 | 23, 18, 12, 8, 32, 35 | syl311anc 1381 | . 2 β’ (π β (βπ₯ β π (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯)) β (πβπΉ) β€ π΄)) |
37 | 31, 36 | mpd 15 | 1 β’ (π β (πβπΉ) β€ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 βwral 3058 class class class wbr 5152 βcfv 6553 (class class class)co 7426 βcr 11145 0cc0 11146 Β· cmul 11151 β€ cle 11287 Basecbs 17187 0gc0g 17428 GrpHom cghm 19174 normcnm 24505 NrmGrpcngp 24506 normOp cnmo 24642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-0g 17430 df-topgen 17432 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-ghm 19175 df-psmet 21278 df-xmet 21279 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 df-nm 24511 df-ngp 24512 df-nmo 24645 |
This theorem is referenced by: nmo0 24672 nmoco 24674 nmotri 24676 nmoid 24679 nmoleub2lem 25061 |
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