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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 6776 | . . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈)) | |
2 | fveq2 6776 | . . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈)) | |
3 | 1, 2 | coeq12d 5775 | . . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
4 | df-ims 28960 | . . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))) | |
5 | fvex 6789 | . . . 4 ⊢ (normCV‘𝑈) ∈ V | |
6 | fvex 6789 | . . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V | |
7 | 5, 6 | coex 7777 | . . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V |
8 | 3, 4, 7 | fvmpt 6877 | . 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 5774 | . 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) |
13 | 8, 9, 12 | 3eqtr4g 2803 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∘ ccom 5595 ‘cfv 6435 NrmCVeccnv 28943 −𝑣 cnsb 28948 normCVcnmcv 28949 IndMetcims 28950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-iota 6393 df-fun 6437 df-fv 6443 df-ims 28960 |
This theorem is referenced by: imsdval 29045 imsdf 29048 cnims 29052 hhims 29531 hhssims 29633 |
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