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Mirrors > Home > MPE Home > Th. List > imsmet | Structured version Visualization version GIF version |
Description: The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsmet.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
imsmet.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsmet | ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsmet.8 | . 2 ⊢ 𝐷 = (IndMet‘𝑈) | |
2 | fveq2 6103 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (IndMet‘𝑈) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
3 | imsmet.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
4 | fveq2 6103 | . . . . . 6 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (BaseSet‘𝑈) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
5 | 3, 4 | syl5eq 2656 | . . . . 5 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → 𝑋 = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
6 | 5 | fveq2d 6107 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → (Met‘𝑋) = (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)))) |
7 | 2, 6 | eleq12d 2682 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) → ((IndMet‘𝑈) ∈ (Met‘𝑋) ↔ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))))) |
8 | eqid 2610 | . . . 4 ⊢ (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
9 | eqid 2610 | . . . 4 ⊢ ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
10 | eqid 2610 | . . . 4 ⊢ (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) = (inv‘( +𝑣 ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) | |
11 | eqid 2610 | . . . 4 ⊢ ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = ( ·𝑠OLD ‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
12 | eqid 2610 | . . . 4 ⊢ (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (0vec‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
13 | eqid 2610 | . . . 4 ⊢ (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (normCV‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
14 | eqid 2610 | . . . 4 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) = (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) | |
15 | elimnvu 26923 | . . . 4 ⊢ if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
16 | 8, 9, 10, 11, 12, 13, 14, 15 | imsmetlem 26929 | . . 3 ⊢ (IndMet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉)) ∈ (Met‘(BaseSet‘if(𝑈 ∈ NrmCVec, 𝑈, 〈〈 + , · 〉, abs〉))) |
17 | 7, 16 | dedth 4089 | . 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) ∈ (Met‘𝑋)) |
18 | 1, 17 | syl5eqel 2692 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 〈cop 4131 ‘cfv 5804 + caddc 9818 · cmul 9820 abscabs 13822 Metcme 19553 invcgn 26729 NrmCVeccnv 26823 +𝑣 cpv 26824 BaseSetcba 26825 ·𝑠OLD cns 26826 0veccn0v 26827 normCVcnmcv 26829 IndMetcims 26830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-met 19561 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 |
This theorem is referenced by: imsxmet 26931 vacn 26933 nmcvcn 26934 smcnlem 26936 blocni 27044 minvecolem2 27115 minvecolem3 27116 minvecolem4a 27117 minvecolem4 27120 minvecolem7 27123 hhmet 27415 hhssmet 27518 |
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