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| Mirrors > Home > MPE Home > Th. List > ncvspds | Structured version Visualization version GIF version | ||
| Description: Value of the distance function in terms of the norm of a normed subcomplex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.) |
| Ref | Expression |
|---|---|
| ncvspds.n | β’ π = (normβπΊ) |
| ncvspds.x | β’ π = (BaseβπΊ) |
| ncvspds.p | β’ + = (+gβπΊ) |
| ncvspds.d | β’ π· = (distβπΊ) |
| ncvspds.s | β’ Β· = ( Β·π βπΊ) |
| Ref | Expression |
|---|---|
| ncvspds | β’ ((πΊ β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ + (-1 Β· π΅)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3965 | . . . 4 β’ (πΊ β (NrmVec β© βVec) β (πΊ β NrmVec β§ πΊ β βVec)) | |
| 2 | nvcnlm 24633 | . . . . . 6 β’ (πΊ β NrmVec β πΊ β NrmMod) | |
| 3 | nlmngp 24614 | . . . . . 6 β’ (πΊ β NrmMod β πΊ β NrmGrp) | |
| 4 | 2, 3 | syl 17 | . . . . 5 β’ (πΊ β NrmVec β πΊ β NrmGrp) |
| 5 | 4 | adantr 479 | . . . 4 β’ ((πΊ β NrmVec β§ πΊ β βVec) β πΊ β NrmGrp) |
| 6 | 1, 5 | sylbi 216 | . . 3 β’ (πΊ β (NrmVec β© βVec) β πΊ β NrmGrp) |
| 7 | ncvspds.n | . . . 4 β’ π = (normβπΊ) | |
| 8 | ncvspds.x | . . . 4 β’ π = (BaseβπΊ) | |
| 9 | eqid 2728 | . . . 4 β’ (-gβπΊ) = (-gβπΊ) | |
| 10 | ncvspds.d | . . . 4 β’ π· = (distβπΊ) | |
| 11 | 7, 8, 9, 10 | ngpds 24533 | . . 3 β’ ((πΊ β NrmGrp β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄(-gβπΊ)π΅))) |
| 12 | 6, 11 | syl3an1 1160 | . 2 β’ ((πΊ β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄(-gβπΊ)π΅))) |
| 13 | id 22 | . . . . . 6 β’ (πΊ β βVec β πΊ β βVec) | |
| 14 | 13 | cvsclm 25073 | . . . . 5 β’ (πΊ β βVec β πΊ β βMod) |
| 15 | 1, 14 | simplbiim 503 | . . . 4 β’ (πΊ β (NrmVec β© βVec) β πΊ β βMod) |
| 16 | ncvspds.p | . . . . 5 β’ + = (+gβπΊ) | |
| 17 | eqid 2728 | . . . . 5 β’ (ScalarβπΊ) = (ScalarβπΊ) | |
| 18 | ncvspds.s | . . . . 5 β’ Β· = ( Β·π βπΊ) | |
| 19 | 8, 16, 9, 17, 18 | clmvsubval 25056 | . . . 4 β’ ((πΊ β βMod β§ π΄ β π β§ π΅ β π) β (π΄(-gβπΊ)π΅) = (π΄ + (-1 Β· π΅))) |
| 20 | 15, 19 | syl3an1 1160 | . . 3 β’ ((πΊ β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΄(-gβπΊ)π΅) = (π΄ + (-1 Β· π΅))) |
| 21 | 20 | fveq2d 6906 | . 2 β’ ((πΊ β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΄(-gβπΊ)π΅)) = (πβ(π΄ + (-1 Β· π΅)))) |
| 22 | 12, 21 | eqtrd 2768 | 1 β’ ((πΊ β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ + (-1 Β· π΅)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3948 βcfv 6553 (class class class)co 7426 1c1 11147 -cneg 11483 Basecbs 17187 +gcplusg 17240 Scalarcsca 17243 Β·π cvsca 17244 distcds 17249 -gcsg 18899 normcnm 24505 NrmGrpcngp 24506 NrmModcnlm 24509 NrmVeccnvc 24510 βModcclm 25009 βVecccvs 25070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-topgen 17432 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cmn 19744 df-mgp 20082 df-ur 20129 df-ring 20182 df-cring 20183 df-subrg 20515 df-lmod 20752 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 df-nm 24511 df-ngp 24512 df-nlm 24515 df-nvc 24516 df-clm 25010 df-cvs 25071 |
| This theorem is referenced by: (None) |
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