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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 | β’ π = ( βπ£ βπ) |
imsval.6 | β’ π = (normCVβπ) |
imsval.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
imsval | β’ (π β NrmCVec β π· = (π β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 β’ (π’ = π β (normCVβπ’) = (normCVβπ)) | |
2 | fveq2 6897 | . . . 4 β’ (π’ = π β ( βπ£ βπ’) = ( βπ£ βπ)) | |
3 | 1, 2 | coeq12d 5867 | . . 3 β’ (π’ = π β ((normCVβπ’) β ( βπ£ βπ’)) = ((normCVβπ) β ( βπ£ βπ))) |
4 | df-ims 30424 | . . 3 β’ IndMet = (π’ β NrmCVec β¦ ((normCVβπ’) β ( βπ£ βπ’))) | |
5 | fvex 6910 | . . . 4 β’ (normCVβπ) β V | |
6 | fvex 6910 | . . . 4 β’ ( βπ£ βπ) β V | |
7 | 5, 6 | coex 7938 | . . 3 β’ ((normCVβπ) β ( βπ£ βπ)) β V |
8 | 3, 4, 7 | fvmpt 7005 | . 2 β’ (π β NrmCVec β (IndMetβπ) = ((normCVβπ) β ( βπ£ βπ))) |
9 | imsval.8 | . 2 β’ π· = (IndMetβπ) | |
10 | imsval.6 | . . 3 β’ π = (normCVβπ) | |
11 | imsval.3 | . . 3 β’ π = ( βπ£ βπ) | |
12 | 10, 11 | coeq12i 5866 | . 2 β’ (π β π) = ((normCVβπ) β ( βπ£ βπ)) |
13 | 8, 9, 12 | 3eqtr4g 2793 | 1 β’ (π β NrmCVec β π· = (π β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β ccom 5682 βcfv 6548 NrmCVeccnv 30407 βπ£ cnsb 30412 normCVcnmcv 30413 IndMetcims 30414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-ims 30424 |
This theorem is referenced by: imsdval 30509 imsdf 30512 cnims 30516 hhims 30995 hhssims 31097 |
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