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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 3967 | . . . 4 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) ↔ (𝐺 ∈ NrmVec ∧ 𝐺 ∈ ℂVec)) | |
| 2 | nvcnlm 24717 | . . . . . 6 ⊢ (𝐺 ∈ NrmVec → 𝐺 ∈ NrmMod) | |
| 3 | nlmngp 24698 | . . . . . 6 ⊢ (𝐺 ∈ NrmMod → 𝐺 ∈ NrmGrp) | |
| 4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ NrmVec → 𝐺 ∈ NrmGrp) |
| 5 | 4 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ NrmVec ∧ 𝐺 ∈ ℂVec) → 𝐺 ∈ NrmGrp) |
| 6 | 1, 5 | sylbi 217 | . . 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 24617 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴(-g‘𝐺)𝐵))) |
| 12 | 6, 11 | syl3an1 1164 | . 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 25159 | . . . . 5 ⊢ (𝐺 ∈ ℂVec → 𝐺 ∈ ℂMod) |
| 15 | 1, 14 | simplbiim 504 | . . . 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 25142 | . . . 4 ⊢ ((𝐺 ∈ ℂMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
| 20 | 15, 19 | syl3an1 1164 | . . 3 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
| 21 | 20 | fveq2d 6910 | . 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ‘cfv 6561 (class class class)co 7431 1c1 11156 -cneg 11493 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑠 cvsca 17301 distcds 17306 -gcsg 18953 normcnm 24589 NrmGrpcngp 24590 NrmModcnlm 24593 NrmVeccnvc 24594 ℂModcclm 25095 ℂVecccvs 25156 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cmn 19800 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-subrg 20570 df-lmod 20860 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nlm 24599 df-nvc 24600 df-clm 25096 df-cvs 25157 |
| This theorem is referenced by: (None) |
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