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Mirrors > Home > MPE Home > Th. List > ncvsprp | Structured version Visualization version GIF version |
Description: Proportionality property of the norm of a scalar product in a normed subcomplex vector space. (Contributed by NM, 11-Nov-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) |
ncvsprp.f | β’ πΉ = (Scalarβπ) |
ncvsprp.k | β’ πΎ = (BaseβπΉ) |
Ref | Expression |
---|---|
ncvsprp | β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β (πβ(π΄ Β· π΅)) = ((absβπ΄) Β· (πβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3963 | . . . 4 β’ (π β (NrmVec β© βVec) β (π β NrmVec β§ π β βVec)) | |
2 | nvcnlm 24631 | . . . . 5 β’ (π β NrmVec β π β NrmMod) | |
3 | 2 | adantr 479 | . . . 4 β’ ((π β NrmVec β§ π β βVec) β π β NrmMod) |
4 | 1, 3 | sylbi 216 | . . 3 β’ (π β (NrmVec β© βVec) β π β NrmMod) |
5 | ncvsprp.v | . . . 4 β’ π = (Baseβπ) | |
6 | ncvsprp.n | . . . 4 β’ π = (normβπ) | |
7 | ncvsprp.s | . . . 4 β’ Β· = ( Β·π βπ) | |
8 | ncvsprp.f | . . . 4 β’ πΉ = (Scalarβπ) | |
9 | ncvsprp.k | . . . 4 β’ πΎ = (BaseβπΉ) | |
10 | eqid 2727 | . . . 4 β’ (normβπΉ) = (normβπΉ) | |
11 | 5, 6, 7, 8, 9, 10 | nmvs 24611 | . . 3 β’ ((π β NrmMod β§ π΄ β πΎ β§ π΅ β π) β (πβ(π΄ Β· π΅)) = (((normβπΉ)βπ΄) Β· (πβπ΅))) |
12 | 4, 11 | syl3an1 1160 | . 2 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β (πβ(π΄ Β· π΅)) = (((normβπΉ)βπ΄) Β· (πβπ΅))) |
13 | id 22 | . . . . . . . 8 β’ (π β βVec β π β βVec) | |
14 | 13 | cvsclm 25071 | . . . . . . 7 β’ (π β βVec β π β βMod) |
15 | 1, 14 | simplbiim 503 | . . . . . 6 β’ (π β (NrmVec β© βVec) β π β βMod) |
16 | 8, 9 | clmabs 25028 | . . . . . 6 β’ ((π β βMod β§ π΄ β πΎ) β (absβπ΄) = ((normβπΉ)βπ΄)) |
17 | 15, 16 | sylan 578 | . . . . 5 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ) β (absβπ΄) = ((normβπΉ)βπ΄)) |
18 | 17 | 3adant3 1129 | . . . 4 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β (absβπ΄) = ((normβπΉ)βπ΄)) |
19 | 18 | eqcomd 2733 | . . 3 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β ((normβπΉ)βπ΄) = (absβπ΄)) |
20 | 19 | oveq1d 7439 | . 2 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β (((normβπΉ)βπ΄) Β· (πβπ΅)) = ((absβπ΄) Β· (πβπ΅))) |
21 | 12, 20 | eqtrd 2767 | 1 β’ ((π β (NrmVec β© βVec) β§ π΄ β πΎ β§ π΅ β π) β (πβ(π΄ Β· π΅)) = ((absβπ΄) Β· (πβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β© cin 3946 βcfv 6551 (class class class)co 7424 Β· cmul 11149 abscabs 15219 Basecbs 17185 Scalarcsca 17241 Β·π cvsca 17242 normcnm 24503 NrmModcnlm 24507 NrmVeccnvc 24508 βModcclm 25007 βVecccvs 25068 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-rp 13013 df-fz 13523 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-0g 17428 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-grp 18898 df-subg 19083 df-cmn 19742 df-mgp 20080 df-ring 20180 df-cring 20181 df-subrg 20513 df-cnfld 21285 df-nm 24509 df-nlm 24513 df-nvc 24514 df-clm 25008 df-cvs 25069 |
This theorem is referenced by: ncvsge0 25099 ncvsm1 25100 ncvspi 25102 |
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