Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6695 | . . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈)) | |
2 | fveq2 6695 | . . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈)) | |
3 | 1, 2 | coeq12d 5718 | . . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
4 | df-ims 28636 | . . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))) | |
5 | fvex 6708 | . . . 4 ⊢ (normCV‘𝑈) ∈ V | |
6 | fvex 6708 | . . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V | |
7 | 5, 6 | coex 7686 | . . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V |
8 | 3, 4, 7 | fvmpt 6796 | . 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 5717 | . 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) |
13 | 8, 9, 12 | 3eqtr4g 2796 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∘ ccom 5540 ‘cfv 6358 NrmCVeccnv 28619 −𝑣 cnsb 28624 normCVcnmcv 28625 IndMetcims 28626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-ims 28636 |
This theorem is referenced by: imsdval 28721 imsdf 28724 cnims 28728 hhims 29207 hhssims 29309 |
Copyright terms: Public domain | W3C validator |