Theorem normval 10623

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 A is usually written as "|| A ||", but we use function value notation to take advantage of our existing theorems about functions.

Assertion

Ref	Expression
normval	|- (A e. ~H -> (normh` A) = (sqr` (A .ih A)))

Proof of Theorem normval

Step	Hyp	Ref	Expression
1		opreq12 4891	. . . 4 |- ((x = A /\ x = A) -> (x .ih x) = (A .ih A))
2	1	anidms 480	. . 3 |- (x = A -> (x .ih x) = (A .ih A))
3	2	fveq2d 4685	. 2 |- (x = A -> (sqr` (x .ih x)) = (sqr` (A .ih A)))
4		dfhnorm2 10621	. 2 |- normh = {<.x, y>. | (x e. ~H /\ y = (sqr` (x .ih x)))}
5		fvex 4689	. 2 |- (sqr` (A .ih A)) e. _V
6	3, 4, 5	fvopab4 4743	1 |- (A e. ~H -> (normh` A) = (sqr` (A .ih A)))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  sqrcsqr 7919  ~Hchil 10420   .ih csp 10425  normhcno 10426

This theorem is referenced by:  normge0 10625  normgt0OLD 10626  normgt0 10627  norm0 10628  normsqi 10632  norm-ii.i 10637  norm-iii.i 10639  bcsiALT 10679

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hfi 10579

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-hnorm 10469

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