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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 6103 | . . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈)) | |
2 | fveq2 6103 | . . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈)) | |
3 | 1, 2 | coeq12d 5208 | . . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
4 | df-ims 26840 | . . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))) | |
5 | fvex 6113 | . . . 4 ⊢ (normCV‘𝑈) ∈ V | |
6 | fvex 6113 | . . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V | |
7 | 5, 6 | coex 7011 | . . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V |
8 | 3, 4, 7 | fvmpt 6191 | . 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 5207 | . 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) |
13 | 8, 9, 12 | 3eqtr4g 2669 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∘ ccom 5042 ‘cfv 5804 NrmCVeccnv 26823 −𝑣 cnsb 26828 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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-ims 26840 |
This theorem is referenced by: imsdval 26925 imsdf 26928 cnims 26932 hhims 27413 hhssims 27516 |
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