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Theorem normval 22583
 Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of is usually written as "|| ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
normval

Proof of Theorem normval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq12 6053 . . . 4
21anidms 627 . . 3
32fveq2d 5695 . 2
4 dfhnorm2 22581 . 2
5 fvex 5705 . 2
63, 4, 5fvmpt 5769 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1649   wcel 1721  cfv 5417  (class class class)co 6044  csqr 11997  chil 22379   csp 22382  cno 22383 This theorem is referenced by:  normge0  22585  normgt0  22586  norm0  22587  normsqi  22591  norm-ii-i  22596  norm-iii-i  22598  bcsiALT  22638 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-hfi 22538 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-hnorm 22428
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