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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 3979 | . . . 4 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) ↔ (𝐺 ∈ NrmVec ∧ 𝐺 ∈ ℂVec)) | |
2 | nvcnlm 24733 | . . . . . 6 ⊢ (𝐺 ∈ NrmVec → 𝐺 ∈ NrmMod) | |
3 | nlmngp 24714 | . . . . . 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 2735 | . . . 4 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
10 | ncvspds.d | . . . 4 ⊢ 𝐷 = (dist‘𝐺) | |
11 | 7, 8, 9, 10 | ngpds 24633 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴(-g‘𝐺)𝐵))) |
12 | 6, 11 | syl3an1 1162 | . 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 25173 | . . . . 5 ⊢ (𝐺 ∈ ℂVec → 𝐺 ∈ ℂMod) |
15 | 1, 14 | simplbiim 504 | . . . 4 ⊢ (𝐺 ∈ (NrmVec ∩ ℂVec) → 𝐺 ∈ ℂMod) |
16 | ncvspds.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
17 | eqid 2735 | . . . . 5 ⊢ (Scalar‘𝐺) = (Scalar‘𝐺) | |
18 | ncvspds.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐺) | |
19 | 8, 16, 9, 17, 18 | clmvsubval 25156 | . . . 4 ⊢ ((𝐺 ∈ ℂMod ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
20 | 15, 19 | syl3an1 1162 | . . 3 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(-g‘𝐺)𝐵) = (𝐴 + (-1 · 𝐵))) |
21 | 20 | fveq2d 6911 | . 2 ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴(-g‘𝐺)𝐵)) = (𝑁‘(𝐴 + (-1 · 𝐵)))) |
22 | 12, 21 | eqtrd 2775 | 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 1086 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ‘cfv 6563 (class class class)co 7431 1c1 11154 -cneg 11491 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑠 cvsca 17302 distcds 17307 -gcsg 18966 normcnm 24605 NrmGrpcngp 24606 NrmModcnlm 24609 NrmVeccnvc 24610 ℂModcclm 25109 ℂVecccvs 25170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-topgen 17490 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-mgp 20153 df-ur 20200 df-ring 20253 df-cring 20254 df-subrg 20587 df-lmod 20877 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-xms 24346 df-ms 24347 df-nm 24611 df-ngp 24612 df-nlm 24615 df-nvc 24616 df-clm 25110 df-cvs 25171 |
This theorem is referenced by: (None) |
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