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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 2726 | . . . 4 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
3 | 1, 2 | nvf 30588 | . . 3 ⊢ (𝑈 ∈ NrmCVec → (normCV‘𝑈):𝑋⟶ℝ) |
4 | eqid 2726 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
5 | 1, 4 | nvmf 30573 | . . 3 ⊢ (𝑈 ∈ NrmCVec → ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) |
6 | fco 6742 | . . 3 ⊢ (((normCV‘𝑈):𝑋⟶ℝ ∧ ( −𝑣 ‘𝑈):(𝑋 × 𝑋)⟶𝑋) → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) | |
7 | 3, 5, 6 | syl2anc 582 | . 2 ⊢ (𝑈 ∈ NrmCVec → ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ) |
8 | imsdfn.8 | . . . 4 ⊢ 𝐷 = (IndMet‘𝑈) | |
9 | 4, 2, 8 | imsval 30613 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
10 | 9 | feq1d 6703 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝐷:(𝑋 × 𝑋)⟶ℝ ↔ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)):(𝑋 × 𝑋)⟶ℝ)) |
11 | 7, 10 | mpbird 256 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 × cxp 5671 ∘ ccom 5677 ⟶wf 6540 ‘cfv 6544 ℝcr 11146 NrmCVeccnv 30512 BaseSetcba 30514 −𝑣 cnsb 30517 normCVcnmcv 30518 IndMetcims 30519 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-ltxr 11292 df-sub 11485 df-neg 11486 df-grpo 30421 df-gid 30422 df-ginv 30423 df-gdiv 30424 df-ablo 30473 df-vc 30487 df-nv 30520 df-va 30523 df-ba 30524 df-sm 30525 df-0v 30526 df-vs 30527 df-nmcv 30528 df-ims 30529 |
This theorem is referenced by: imsmetlem 30618 sspims 30667 |
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