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| Mirrors > Home > MPE Home > Th. List > nmcvfval | Structured version Visualization version GIF version | ||
| Description: Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nmfval.6 | ⊢ 𝑁 = (normCV‘𝑈) | 
| Ref | Expression | 
|---|---|
| nmcvfval | ⊢ 𝑁 = (2nd ‘𝑈) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nmfval.6 | . 2 ⊢ 𝑁 = (normCV‘𝑈) | |
| 2 | df-nmcv 30620 | . . 3 ⊢ normCV = 2nd | |
| 3 | 2 | fveq1i 6906 | . 2 ⊢ (normCV‘𝑈) = (2nd ‘𝑈) | 
| 4 | 1, 3 | eqtri 2764 | 1 ⊢ 𝑁 = (2nd ‘𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ‘cfv 6560 2nd c2nd 8014 normCVcnmcv 30610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-nmcv 30620 | 
| This theorem is referenced by: nvop2 30628 nvop 30696 cnnvnm 30701 phop 30838 h2hnm 30996 hhssnm 31279 | 
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