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Mirrors > Home > MPE Home > Th. List > nmoge0 | Structured version Visualization version GIF version |
Description: The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | β’ π = (π normOp π) |
Ref | Expression |
---|---|
nmoge0 | β’ ((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β 0 β€ (πβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrege0 13435 | . . . . . 6 β’ (π β (0[,)+β) β (π β β β§ 0 β€ π)) | |
2 | 1 | simprbi 497 | . . . . 5 β’ (π β (0[,)+β) β 0 β€ π) |
3 | 2 | adantl 482 | . . . 4 β’ (((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β§ π β (0[,)+β)) β 0 β€ π) |
4 | 3 | a1d 25 | . . 3 β’ (((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β§ π β (0[,)+β)) β (βπ₯ β (Baseβπ)((normβπ)β(πΉβπ₯)) β€ (π Β· ((normβπ)βπ₯)) β 0 β€ π)) |
5 | 4 | ralrimiva 3146 | . 2 β’ ((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β βπ β (0[,)+β)(βπ₯ β (Baseβπ)((normβπ)β(πΉβπ₯)) β€ (π Β· ((normβπ)βπ₯)) β 0 β€ π)) |
6 | 0xr 11265 | . . 3 β’ 0 β β* | |
7 | nmofval.1 | . . . 4 β’ π = (π normOp π) | |
8 | eqid 2732 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
9 | eqid 2732 | . . . 4 β’ (normβπ) = (normβπ) | |
10 | eqid 2732 | . . . 4 β’ (normβπ) = (normβπ) | |
11 | 7, 8, 9, 10 | nmogelb 24453 | . . 3 β’ (((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β§ 0 β β*) β (0 β€ (πβπΉ) β βπ β (0[,)+β)(βπ₯ β (Baseβπ)((normβπ)β(πΉβπ₯)) β€ (π Β· ((normβπ)βπ₯)) β 0 β€ π))) |
12 | 6, 11 | mpan2 689 | . 2 β’ ((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β (0 β€ (πβπΉ) β βπ β (0[,)+β)(βπ₯ β (Baseβπ)((normβπ)β(πΉβπ₯)) β€ (π Β· ((normβπ)βπ₯)) β 0 β€ π))) |
13 | 5, 12 | mpbird 256 | 1 β’ ((π β NrmGrp β§ π β NrmGrp β§ πΉ β (π GrpHom π)) β 0 β€ (πβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 class class class wbr 5148 βcfv 6543 (class class class)co 7411 βcr 11111 0cc0 11112 Β· cmul 11117 +βcpnf 11249 β*cxr 11251 β€ cle 11253 [,)cico 13330 Basecbs 17148 GrpHom cghm 19127 normcnm 24305 NrmGrpcngp 24306 normOp cnmo 24442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-ico 13334 df-nmo 24445 |
This theorem is referenced by: isnghm3 24462 bddnghm 24463 nmoi 24465 nmoix 24466 nmo0 24472 nmoco 24474 nmotri 24476 nmoid 24479 nghmcn 24482 nmoleub2lem 24854 |
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