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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 24638 | . 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 2728 | . . . 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 4075 | . . . . . 6 β’ {π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))} β (0[,)+β) | |
4 | icossxr 13442 | . . . . . 6 β’ (0[,)+β) β β* | |
5 | 3, 4 | sstri 3989 | . . . . 5 β’ {π β (0[,)+β) β£ βπ₯ β (Baseβπ )((normβπ‘)β(πβπ₯)) β€ (π Β· ((normβπ )βπ₯))} β β* |
6 | infxrcl 13345 | . . . . 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 7122 | . . 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 7453 | . . 3 β’ (π GrpHom π‘) β V | |
10 | xrex 13002 | . . 3 β’ β* β V | |
11 | fex2 7941 | . . 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 1458 | . 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 8074 | 1 β’ normOp Fn (NrmGrp Γ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2099 βwral 3058 {crab 3429 Vcvv 3471 β wss 3947 class class class wbr 5148 β¦ cmpt 5231 Γ cxp 5676 Fn wfn 6543 βΆwf 6544 βcfv 6548 (class class class)co 7420 infcinf 9465 0cc0 11139 Β· cmul 11144 +βcpnf 11276 β*cxr 11278 < clt 11279 β€ cle 11280 [,)cico 13359 Basecbs 17180 GrpHom cghm 19167 normcnm 24498 NrmGrpcngp 24499 normOp cnmo 24635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-ico 13363 df-nmo 24638 |
This theorem is referenced by: nghmfval 24652 isnghm 24653 |
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