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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 2732 | . . . . . 6 β’ (π· βΎ (π Γ π)) = (π· βΎ (π Γ π)) | |
6 | 1, 2, 3, 4, 5 | isngp2 24105 | . . . . 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 1147 | . . . 4 β’ (πΊ β NrmGrp β (π β β ) = (π· βΎ (π Γ π))) |
8 | 7 | 3ad2ant1 1133 | . . 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 7425 | . 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 24107 | . . . . . 6 β’ (πΊ β NrmGrp β πΊ β Grp) | |
11 | 4, 2 | grpsubf 18901 | . . . . . 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 1133 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β β :(π Γ π)βΆπ) |
14 | opelxpi 5713 | . . . . 5 β’ ((π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) | |
15 | 14 | 3adant1 1130 | . . . 4 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) |
16 | fvco3 6990 | . . . 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 584 | . . 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 7411 | . . 3 β’ (π΄(π β β )π΅) = ((π β β )ββ¨π΄, π΅β©) | |
19 | df-ov 7411 | . . . 4 β’ (π΄ β π΅) = ( β ββ¨π΄, π΅β©) | |
20 | 19 | fveq2i 6894 | . . 3 β’ (πβ(π΄ β π΅)) = (πβ( β ββ¨π΄, π΅β©)) |
21 | 17, 18, 20 | 3eqtr4g 2797 | . 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 7572 | . . 3 β’ ((π΄ β π β§ π΅ β π) β (π΄(π· βΎ (π Γ π))π΅) = (π΄π·π΅)) | |
23 | 22 | 3adant1 1130 | . 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 2781 | 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 1087 = wceq 1541 β wcel 2106 β¨cop 4634 Γ cxp 5674 βΎ cres 5678 β ccom 5680 βΆwf 6539 βcfv 6543 (class class class)co 7408 Basecbs 17143 distcds 17205 Grpcgrp 18818 -gcsg 18820 MetSpcms 23823 normcnm 24084 NrmGrpcngp 24085 |
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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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-pss 3967 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-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-n0 12472 df-z 12558 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-0g 17386 df-topgen 17388 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-sbg 18823 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-xms 23825 df-ms 23826 df-nm 24090 df-ngp 24091 |
This theorem is referenced by: ngpdsr 24113 ngpds2 24114 ngprcan 24118 ngpinvds 24121 nmmtri 24130 nmrtri 24132 subgngp 24143 nrgdsdi 24181 nrgdsdir 24182 nlmdsdi 24197 nlmdsdir 24198 nrginvrcnlem 24207 nmods 24260 ncvspds 24677 ipcnlem2 24760 minveclem2 24942 minveclem3b 24944 minveclem4 24948 minveclem6 24950 |
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