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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 2760 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
3 | 1, 2 | nvf 27845 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (normCV‘𝑈):𝑋⟶ℝ) |
4 | eqid 2760 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
5 | 1, 4 | nvmf 27830 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) |
6 | fco 6219 | . . 3 ⊢ (((normCV‘𝑈):𝑋⟶ℝ ∧ ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) | |
7 | 3, 5, 6 | syl2anc 696 | . 2 ⊢ (𝑈 ∈ NrmCVec → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) |
8 | imsdfn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
9 | 4, 2, 8 | imsval 27870 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
10 | 9 | feq1d 6191 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ)) |
11 | 7, 10 | mpbird 247 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 × cxp 5264 ∘ ccom 5270 ⟶wf 6045 ‘cfv 6049 ℝcr 10147 NrmCVeccnv 27769 BaseSetcba 27771 −𝑣 cnsb 27774 normCVcnmcv 27775 IndMetcims 27776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-1st 7334 df-2nd 7335 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-ltxr 10291 df-sub 10480 df-neg 10481 df-grpo 27677 df-gid 27678 df-ginv 27679 df-gdiv 27680 df-ablo 27729 df-vc 27744 df-nv 27777 df-va 27780 df-ba 27781 df-sm 27782 df-0v 27783 df-vs 27784 df-nmcv 27785 df-ims 27786 |
This theorem is referenced by: imsmetlem 27875 sspims 27924 |
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