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Mirrors > Home > MPE Home > Th. List > nmoffn | Structured version Visualization version GIF version |
Description: The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
nmoffn | β’ normOp Fn (NrmGrp Γ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nmo 24095 | . 2 β’ normOp = (π β NrmGrp, π‘ β NrmGrp β¦ (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < ))) | |
2 | eqid 2733 | . . . 4 β’ (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )) = (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )) | |
3 | ssrab2 4041 | . . . . . 6 β’ {π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))} β (0[,)+β) | |
4 | icossxr 13358 | . . . . . 6 β’ (0[,)+β) β β* | |
5 | 3, 4 | sstri 3957 | . . . . 5 β’ {π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))} β β* |
6 | infxrcl 13261 | . . . . 5 β’ ({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))} β β* β inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < ) β β*) | |
7 | 5, 6 | mp1i 13 | . . . 4 β’ (π β (π GrpHom π‘) β inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < ) β β*) |
8 | 2, 7 | fmpti 7064 | . . 3 β’ (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )):(π GrpHom π‘)βΆβ* |
9 | ovex 7394 | . . 3 β’ (π GrpHom π‘) β V | |
10 | xrex 12920 | . . 3 β’ β* β V | |
11 | fex2 7874 | . . 3 β’ (((π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )):(π GrpHom π‘)βΆβ* β§ (π GrpHom π‘) β V β§ β* β V) β (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )) β V) | |
12 | 8, 9, 10, 11 | mp3an 1462 | . 2 β’ (π β (π GrpHom π‘) β¦ inf({π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))}, β*, < )) β V |
13 | 1, 12 | fnmpoi 8006 | 1 β’ normOp Fn (NrmGrp Γ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2107 βwral 3061 {crab 3406 Vcvv 3447 β wss 3914 class class class wbr 5109 β¦ cmpt 5192 Γ cxp 5635 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 infcinf 9385 0cc0 11059 Β· cmul 11064 +βcpnf 11194 β*cxr 11196 < clt 11197 β€ cle 11198 [,)cico 13275 Basecbs 17091 GrpHom cghm 19013 normcnm 23955 NrmGrpcngp 23956 normOp cnmo 24092 |
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-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-ico 13279 df-nmo 24095 |
This theorem is referenced by: nghmfval 24109 isnghm 24110 |
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