![]() |
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 6493 | . . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈)) | |
2 | fveq2 6493 | . . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈)) | |
3 | 1, 2 | coeq12d 5578 | . . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))) |
4 | df-ims 28145 | . . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))) | |
5 | fvex 6506 | . . . 4 ⊢ (normCV‘𝑈) ∈ V | |
6 | fvex 6506 | . . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V | |
7 | 5, 6 | coex 7444 | . . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V |
8 | 3, 4, 7 | fvmpt 6589 | . 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 5577 | . 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) |
13 | 8, 9, 12 | 3eqtr4g 2833 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 ∘ ccom 5404 ‘cfv 6182 NrmCVeccnv 28128 −𝑣 cnsb 28133 normCVcnmcv 28134 IndMetcims 28135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-iota 6146 df-fun 6184 df-fv 6190 df-ims 28145 |
This theorem is referenced by: imsdval 28230 imsdf 28233 cnims 28237 hhims 28718 hhssims 28821 |
Copyright terms: Public domain | W3C validator |