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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 | ⊢ 𝑁 = (norm‘𝐺) |
ncvspds.x | ⊢ 𝑋 = (Base‘𝐺) |
ncvspds.p | ⊢ + = (+g‘𝐺) |
ncvspds.d | ⊢ 𝐷 = (dist‘𝐺) |
ncvspds.s | ⊢ · = ( ·𝑠 ‘𝐺) |
Ref | Expression |
---|---|
ncvspds | ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3924 | . . . 4 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) ↔ (𝐺 ∈ NrmVec ∧ 𝐺 ∈ ℂVec)) | |
2 | nvcnlm 24011 | . . . . . 6 ⊢ (𝐺 ∈ NrmVec → 𝐺 ∈ NrmMod) | |
3 | nlmngp 23992 | . . . . . 6 ⊢ (𝐺 ∈ NrmMod → 𝐺 ∈ NrmGrp) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmVec → 𝐺 ∈ NrmGrp) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝐺 ∈ NrmVec ∧ 𝐺 ∈ ℂVec) → 𝐺 ∈ NrmGrp) |
6 | 1, 5 | sylbi 216 | . . 3 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) → 𝐺 ∈ NrmGrp) |
7 | ncvspds.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
8 | ncvspds.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
9 | eqid 2737 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
10 | ncvspds.d | . . . 4 ⊢ 𝐷 = (dist‘𝐺) | |
11 | 7, 8, 9, 10 | ngpds 23911 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴(-g‘𝐺)𝐵))) |
12 | 6, 11 | syl3an1 1163 | . 2 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴(-g‘𝐺)𝐵))) |
13 | id 22 | . . . . . 6 ⊢ (𝐺 ∈ ℂVec → 𝐺 ∈ ℂVec) | |
14 | 13 | cvsclm 24440 | . . . . 5 ⊢ (𝐺 ∈ ℂVec → 𝐺 ∈ ℂMod) |
15 | 1, 14 | simplbiim 505 | . . . 4 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) → 𝐺 ∈ ℂMod) |
16 | ncvspds.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
17 | eqid 2737 | . . . . 5 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
18 | ncvspds.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐺) | |
19 | 8, 16, 9, 17, 18 | clmvsubval 24423 | . . . 4 ⊢ ((𝐺 ∈ ℂMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
20 | 15, 19 | syl3an1 1163 | . . 3 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
21 | 20 | fveq2d 6843 | . 2 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴(-g‘𝐺)𝐵)) = (𝑁‘(𝐴 + (-1 · 𝐵)))) |
22 | 12, 21 | eqtrd 2777 | 1 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3907 ‘cfv 6493 (class class class)co 7351 1c1 11010 -cneg 11344 Basecbs 17042 +gcplusg 17092 Scalarcsca 17095 ·𝑠 cvsca 17096 distcds 17101 -gcsg 18709 normcnm 23883 NrmGrpcngp 23884 NrmModcnlm 23887 NrmVeccnvc 23888 ℂModcclm 24376 ℂVecccvs 24437 |
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 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-fz 13379 df-struct 16978 df-sets 16995 df-slot 17013 df-ndx 17025 df-base 17043 df-ress 17072 df-plusg 17105 df-mulr 17106 df-starv 17107 df-tset 17111 df-ple 17112 df-ds 17114 df-unif 17115 df-0g 17282 df-topgen 17284 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 df-minusg 18711 df-sbg 18712 df-subg 18883 df-cmn 19522 df-mgp 19855 df-ur 19872 df-ring 19919 df-cring 19920 df-subrg 20172 df-lmod 20276 df-psmet 20740 df-xmet 20741 df-met 20742 df-bl 20743 df-mopn 20744 df-cnfld 20749 df-top 22194 df-topon 22211 df-topsp 22233 df-bases 22247 df-xms 23624 df-ms 23625 df-nm 23889 df-ngp 23890 df-nlm 23893 df-nvc 23894 df-clm 24377 df-cvs 24438 |
This theorem is referenced by: (None) |
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