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Mirrors > Home > MPE Home > Th. List > ngpds | Structured version Visualization version GIF version |
Description: Value of the distance function in terms of the norm of a normed group. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
ngpds.n | β’ π = (normβπΊ) |
ngpds.x | β’ π = (BaseβπΊ) |
ngpds.m | β’ β = (-gβπΊ) |
ngpds.d | β’ π· = (distβπΊ) |
Ref | Expression |
---|---|
ngpds | β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ β π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpds.n | . . . . . 6 β’ π = (normβπΊ) | |
2 | ngpds.m | . . . . . 6 β’ β = (-gβπΊ) | |
3 | ngpds.d | . . . . . 6 β’ π· = (distβπΊ) | |
4 | ngpds.x | . . . . . 6 β’ π = (BaseβπΊ) | |
5 | eqid 2733 | . . . . . 6 β’ (π· βΎ (π Γ π)) = (π· βΎ (π Γ π)) | |
6 | 1, 2, 3, 4, 5 | isngp2 23976 | . . . . 5 β’ (πΊ β NrmGrp β (πΊ β Grp β§ πΊ β MetSp β§ (π β β ) = (π· βΎ (π Γ π)))) |
7 | 6 | simp3bi 1148 | . . . 4 β’ (πΊ β NrmGrp β (π β β ) = (π· βΎ (π Γ π))) |
8 | 7 | 3ad2ant1 1134 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π β β ) = (π· βΎ (π Γ π))) |
9 | 8 | oveqd 7378 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄(π β β )π΅) = (π΄(π· βΎ (π Γ π))π΅)) |
10 | ngpgrp 23978 | . . . . . 6 β’ (πΊ β NrmGrp β πΊ β Grp) | |
11 | 4, 2 | grpsubf 18834 | . . . . . 6 β’ (πΊ β Grp β β :(π Γ π)βΆπ) |
12 | 10, 11 | syl 17 | . . . . 5 β’ (πΊ β NrmGrp β β :(π Γ π)βΆπ) |
13 | 12 | 3ad2ant1 1134 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β β :(π Γ π)βΆπ) |
14 | opelxpi 5674 | . . . . 5 β’ ((π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) | |
15 | 14 | 3adant1 1131 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) |
16 | fvco3 6944 | . . . 4 β’ (( β :(π Γ π)βΆπ β§ β¨π΄, π΅β© β (π Γ π)) β ((π β β )ββ¨π΄, π΅β©) = (πβ( β ββ¨π΄, π΅β©))) | |
17 | 13, 15, 16 | syl2anc 585 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β ((π β β )ββ¨π΄, π΅β©) = (πβ( β ββ¨π΄, π΅β©))) |
18 | df-ov 7364 | . . 3 β’ (π΄(π β β )π΅) = ((π β β )ββ¨π΄, π΅β©) | |
19 | df-ov 7364 | . . . 4 β’ (π΄ β π΅) = ( β ββ¨π΄, π΅β©) | |
20 | 19 | fveq2i 6849 | . . 3 β’ (πβ(π΄ β π΅)) = (πβ( β ββ¨π΄, π΅β©)) |
21 | 17, 18, 20 | 3eqtr4g 2798 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄(π β β )π΅) = (πβ(π΄ β π΅))) |
22 | ovres 7524 | . . 3 β’ ((π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) | |
23 | 22 | 3adant1 1131 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) |
24 | 9, 21, 23 | 3eqtr3rd 2782 | 1 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ β π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4596 Γ cxp 5635 βΎ cres 5639 β ccom 5641 βΆwf 6496 βcfv 6500 (class class class)co 7361 Basecbs 17091 distcds 17150 Grpcgrp 18756 -gcsg 18758 MetSpcms 23694 normcnm 23955 NrmGrpcngp 23956 |
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-pss 3933 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-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 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-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 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-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 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-div 11821 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-0g 17331 df-topgen 17333 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-sbg 18761 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-xms 23696 df-ms 23697 df-nm 23961 df-ngp 23962 |
This theorem is referenced by: ngpdsr 23984 ngpds2 23985 ngprcan 23989 ngpinvds 23992 nmmtri 24001 nmrtri 24003 subgngp 24014 nrgdsdi 24052 nrgdsdir 24053 nlmdsdi 24068 nlmdsdir 24069 nrginvrcnlem 24078 nmods 24131 ncvspds 24548 ipcnlem2 24631 minveclem2 24813 minveclem3b 24815 minveclem4 24819 minveclem6 24821 |
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