![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imsdf | Structured version Visualization version GIF version |
Description: Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdfn.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
imsdfn.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsdf | ⊢ (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdfn.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | eqid 2777 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
3 | 1, 2 | nvf 28087 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (normCV‘𝑈):𝑋⟶ℝ) |
4 | eqid 2777 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
5 | 1, 4 | nvmf 28072 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) |
6 | fco 6308 | . . 3 ⊢ (((normCV‘𝑈):𝑋⟶ℝ ∧ ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) | |
7 | 3, 5, 6 | syl2anc 579 | . 2 ⊢ (𝑈 ∈ NrmCVec → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) |
8 | imsdfn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
9 | 4, 2, 8 | imsval 28112 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
10 | 9 | feq1d 6276 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ)) |
11 | 7, 10 | mpbird 249 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 × cxp 5353 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 ℝcr 10271 NrmCVeccnv 28011 BaseSetcba 28013 −𝑣 cnsb 28016 normCVcnmcv 28017 IndMetcims 28018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-neg 10609 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 |
This theorem is referenced by: imsmetlem 28117 sspims 28166 |
Copyright terms: Public domain | W3C validator |