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Mirrors > Home > MPE Home > Th. List > ncvsdif | Structured version Visualization version GIF version |
Description: The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.) |
Ref | Expression |
---|---|
ncvsprp.v | β’ π = (Baseβπ) |
ncvsprp.n | β’ π = (normβπ) |
ncvsprp.s | β’ Β· = ( Β·π βπ) |
ncvsdif.p | β’ + = (+gβπ) |
Ref | Expression |
---|---|
ncvsdif | β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΄ + (-1 Β· π΅))) = (πβ(π΅ + (-1 Β· π΄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3965 | . . . . 5 β’ (π β (NrmVec β© βVec) β (π β NrmVec β§ π β βVec)) | |
2 | id 22 | . . . . . 6 β’ (π β βVec β π β βVec) | |
3 | 2 | cvsclm 25073 | . . . . 5 β’ (π β βVec β π β βMod) |
4 | 1, 3 | simplbiim 503 | . . . 4 β’ (π β (NrmVec β© βVec) β π β βMod) |
5 | ncvsprp.v | . . . . . 6 β’ π = (Baseβπ) | |
6 | ncvsdif.p | . . . . . 6 β’ + = (+gβπ) | |
7 | eqid 2728 | . . . . . 6 β’ (-gβπ) = (-gβπ) | |
8 | eqid 2728 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
9 | ncvsprp.s | . . . . . 6 β’ Β· = ( Β·π βπ) | |
10 | 5, 6, 7, 8, 9 | clmvsubval 25056 | . . . . 5 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄(-gβπ)π΅) = (π΄ + (-1 Β· π΅))) |
11 | 10 | eqcomd 2734 | . . . 4 β’ ((π β βMod β§ π΄ β π β§ π΅ β π) β (π΄ + (-1 Β· π΅)) = (π΄(-gβπ)π΅)) |
12 | 4, 11 | syl3an1 1160 | . . 3 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΄ + (-1 Β· π΅)) = (π΄(-gβπ)π΅)) |
13 | 12 | fveq2d 6906 | . 2 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΄ + (-1 Β· π΅))) = (πβ(π΄(-gβπ)π΅))) |
14 | nvcnlm 24633 | . . . . . 6 β’ (π β NrmVec β π β NrmMod) | |
15 | nlmngp 24614 | . . . . . 6 β’ (π β NrmMod β π β NrmGrp) | |
16 | 14, 15 | syl 17 | . . . . 5 β’ (π β NrmVec β π β NrmGrp) |
17 | 16 | adantr 479 | . . . 4 β’ ((π β NrmVec β§ π β βVec) β π β NrmGrp) |
18 | 1, 17 | sylbi 216 | . . 3 β’ (π β (NrmVec β© βVec) β π β NrmGrp) |
19 | ncvsprp.n | . . . 4 β’ π = (normβπ) | |
20 | 5, 19, 7 | nmsub 24552 | . . 3 β’ ((π β NrmGrp β§ π΄ β π β§ π΅ β π) β (πβ(π΄(-gβπ)π΅)) = (πβ(π΅(-gβπ)π΄))) |
21 | 18, 20 | syl3an1 1160 | . 2 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΄(-gβπ)π΅)) = (πβ(π΅(-gβπ)π΄))) |
22 | 4 | 3ad2ant1 1130 | . . . 4 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β π β βMod) |
23 | simp3 1135 | . . . 4 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β π΅ β π) | |
24 | simp2 1134 | . . . 4 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β π΄ β π) | |
25 | 5, 6, 7, 8, 9 | clmvsubval 25056 | . . . 4 β’ ((π β βMod β§ π΅ β π β§ π΄ β π) β (π΅(-gβπ)π΄) = (π΅ + (-1 Β· π΄))) |
26 | 22, 23, 24, 25 | syl3anc 1368 | . . 3 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (π΅(-gβπ)π΄) = (π΅ + (-1 Β· π΄))) |
27 | 26 | fveq2d 6906 | . 2 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΅(-gβπ)π΄)) = (πβ(π΅ + (-1 Β· π΄)))) |
28 | 13, 21, 27 | 3eqtrd 2772 | 1 β’ ((π β (NrmVec β© βVec) β§ π΄ β π β§ π΅ β π) β (πβ(π΄ + (-1 Β· 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 -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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