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Mirrors > Home > MPE Home > Th. List > imsdval2 | Structured version Visualization version GIF version |
Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdval2.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
imsdval2.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
imsdval2.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
imsdval2.6 | ⊢ 𝑁 = (normCV‘𝑈) |
imsdval2.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsdval2 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdval2.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | eqid 2825 | . . 3 ⊢ ( −𝑣 ‘𝑈) = ( −𝑣 ‘𝑈) | |
3 | imsdval2.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
4 | imsdval2.8 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
5 | 1, 2, 3, 4 | imsdval 28096 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴( −𝑣 ‘𝑈)𝐵))) |
6 | imsdval2.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
7 | imsdval2.4 | . . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
8 | 1, 6, 7, 2 | nvmval 28052 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( −𝑣 ‘𝑈)𝐵) = (𝐴𝐺(-1𝑆𝐵))) |
9 | 8 | fveq2d 6437 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴( −𝑣 ‘𝑈)𝐵)) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) |
10 | 5, 9 | eqtrd 2861 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 1c1 10253 -cneg 10586 NrmCVeccnv 27994 +𝑣 cpv 27995 BaseSetcba 27996 ·𝑠OLD cns 27997 −𝑣 cnsb 27999 normCVcnmcv 28000 IndMetcims 28001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-po 5263 df-so 5264 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-ltxr 10396 df-sub 10587 df-neg 10588 df-grpo 27903 df-gid 27904 df-ginv 27905 df-gdiv 27906 df-ablo 27955 df-vc 27969 df-nv 28002 df-va 28005 df-ba 28006 df-sm 28007 df-0v 28008 df-vs 28009 df-nmcv 28010 df-ims 28011 |
This theorem is referenced by: imsmetlem 28100 nmcvcn 28105 smcnlem 28107 |
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