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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 24326 | . . . . 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 7428 | . 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 24328 | . . . . . 6 β’ (πΊ β NrmGrp β πΊ β Grp) | |
11 | 4, 2 | grpsubf 18938 | . . . . . 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 7414 | . . 3 β’ (π΄(π β β )π΅) = ((π β β )ββ¨π΄, π΅β©) | |
19 | df-ov 7414 | . . . 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 7575 | . . 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 7411 Basecbs 17148 distcds 17210 Grpcgrp 18855 -gcsg 18857 MetSpcms 24044 normcnm 24305 NrmGrpcngp 24306 |
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-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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 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-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-0g 17391 df-topgen 17393 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-xms 24046 df-ms 24047 df-nm 24311 df-ngp 24312 |
This theorem is referenced by: ngpdsr 24334 ngpds2 24335 ngprcan 24339 ngpinvds 24342 nmmtri 24351 nmrtri 24353 subgngp 24364 nrgdsdi 24402 nrgdsdir 24403 nlmdsdi 24418 nlmdsdir 24419 nrginvrcnlem 24428 nmods 24481 ncvspds 24902 ipcnlem2 24985 minveclem2 25167 minveclem3b 25169 minveclem4 25173 minveclem6 25175 |
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