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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 6848 | . . . . 5 β’ (π₯ = 0 β (πβ(πΉβπ₯)) = (πβ(πΉβ 0 ))) | |
2 | fveq2 6843 | . . . . . 6 β’ (π₯ = 0 β (πΏβπ₯) = (πΏβ 0 )) | |
3 | 2 | oveq2d 7374 | . . . . 5 β’ (π₯ = 0 β (π΄ Β· (πΏβπ₯)) = (π΄ Β· (πΏβ 0 ))) |
4 | 1, 3 | breq12d 5119 | . . . 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 469 | . . . 4 β’ (((π β§ π₯ β π) β§ π₯ β 0 ) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
7 | 0le0 12259 | . . . . . . 7 β’ 0 β€ 0 | |
8 | nmolb2d.4 | . . . . . . . . 9 β’ (π β π΄ β β) | |
9 | 8 | recnd 11188 | . . . . . . . 8 β’ (π β π΄ β β) |
10 | 9 | mul01d 11359 | . . . . . . 7 β’ (π β (π΄ Β· 0) = 0) |
11 | 7, 10 | breqtrrid 5144 | . . . . . 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 2733 | . . . . . . . . . 10 β’ (0gβπ) = (0gβπ) | |
15 | 13, 14 | ghmid 19019 | . . . . . . . . 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 6847 | . . . . . . 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 24001 | . . . . . . . 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 2773 | . . . . . 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 24001 | . . . . . . . 8 β’ (π β NrmGrp β (πΏβ 0 ) = 0) |
26 | 23, 25 | syl 17 | . . . . . . 7 β’ (π β (πΏβ 0 ) = 0) |
27 | 26 | oveq2d 7374 | . . . . . 6 β’ (π β (π΄ Β· (πΏβ 0 )) = (π΄ Β· 0)) |
28 | 11, 22, 27 | 3brtr4d 5138 | . . . . 5 β’ (π β (πβ(πΉβ 0 )) β€ (π΄ Β· (πΏβ 0 ))) |
29 | 28 | adantr 482 | . . . 4 β’ ((π β§ π₯ β π) β (πβ(πΉβ 0 )) β€ (π΄ Β· (πΏβ 0 ))) |
30 | 4, 6, 29 | pm2.61ne 3027 | . . 3 β’ ((π β§ π₯ β π) β (πβ(πΉβπ₯)) β€ (π΄ Β· (πΏβπ₯))) |
31 | 30 | ralrimiva 3140 | . 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 24097 | . . 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 1385 | . 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 397 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 class class class wbr 5106 βcfv 6497 (class class class)co 7358 βcr 11055 0cc0 11056 Β· cmul 11061 β€ cle 11195 Basecbs 17088 0gc0g 17326 GrpHom cghm 19010 normcnm 23948 NrmGrpcngp 23949 normOp cnmo 24085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13276 df-0g 17328 df-topgen 17330 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-ghm 19011 df-psmet 20804 df-xmet 20805 df-bl 20807 df-mopn 20808 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-xms 23689 df-ms 23690 df-nm 23954 df-ngp 23955 df-nmo 24088 |
This theorem is referenced by: nmo0 24115 nmoco 24117 nmotri 24119 nmoid 24122 nmoleub2lem 24493 |
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