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Mirrors > Home > MPE Home > Th. List > ngpdsr | Structured version Visualization version GIF version |
Description: Value of the distance function in terms of the norm of a normed group. (Contributed 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 |
---|---|
ngpdsr | 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 | ngpxms 24503 | . . 3 β’ (πΊ β NrmGrp β πΊ β βMetSp) | |
2 | ngpds.x | . . . 4 β’ π = (BaseβπΊ) | |
3 | ngpds.d | . . . 4 β’ π· = (distβπΊ) | |
4 | 2, 3 | xmssym 24364 | . . 3 β’ ((πΊ β βMetSp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (π΅π·π΄)) |
5 | 1, 4 | syl3an1 1161 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (π΅π·π΄)) |
6 | ngpds.n | . . . 4 β’ π = (normβπΊ) | |
7 | ngpds.m | . . . 4 β’ β = (-gβπΊ) | |
8 | 6, 2, 7, 3 | ngpds 24506 | . . 3 β’ ((πΊ β NrmGrp β§ π΅ β π β§ π΄ β π) β (π΅π·π΄) = (πβ(π΅ β π΄))) |
9 | 8 | 3com23 1124 | . 2 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΅π·π΄) = (πβ(π΅ β π΄))) |
10 | 5, 9 | eqtrd 2768 | 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 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 Basecbs 17173 distcds 17235 -gcsg 18885 βMetSpcxms 24216 normcnm 24478 NrmGrpcngp 24479 |
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 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-0g 17416 df-topgen 17418 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-xms 24219 df-ms 24220 df-nm 24484 df-ngp 24485 |
This theorem is referenced by: ngpinvds 24515 nminv 24523 nlmvscnlem2 24595 nrginvrcnlem 24601 ngpocelbl 24614 ipcnlem2 25165 minveclem2 25347 qqhcn 33586 qqhucn 33587 |
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