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Mirrors > Home > MPE Home > Th. List > imsval | Structured version Visualization version GIF version |
Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
imsval.6 | ⊢ 𝑁 = (normCV‘𝑈) |
imsval.8 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
imsval | ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈)) | |
2 | fveq2 6664 | . . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈)) | |
3 | 1, 2 | coeq12d 5729 | . . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
4 | df-ims 28372 | . . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))) | |
5 | fvex 6677 | . . . 4 ⊢ (normCV‘𝑈) ∈ V | |
6 | fvex 6677 | . . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V | |
7 | 5, 6 | coex 7629 | . . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V |
8 | 3, 4, 7 | fvmpt 6762 | . 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
9 | imsval.8 | . 2 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | imsval.6 | . . 3 ⊢ 𝑁 = (normCV‘𝑈) | |
11 | imsval.3 | . . 3 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
12 | 10, 11 | coeq12i 5728 | . 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) |
13 | 8, 9, 12 | 3eqtr4g 2881 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∘ ccom 5553 ‘cfv 6349 NrmCVeccnv 28355 −𝑣 cnsb 28360 normCVcnmcv 28361 IndMetcims 28362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-ims 28372 |
This theorem is referenced by: imsdval 28457 imsdf 28460 cnims 28464 hhims 28943 hhssims 29045 |
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