Theorem normvalt 8990

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
normvalt	|- (A e. H~ -> (normh` A) = (sqr` (A .ih A)))

Proof of Theorem normvalt

Step	Hyp	Ref	Expression
1		opreq12 3970	. . . 4 |- ((x = A /\ x = A) -> (x .ih x) = (A .ih A))
2	1	anidms 434	. . 3 |- (x = A -> (x .ih x) = (A .ih A))
3	2	fveq2d 3728	. 2 |- (x = A -> (sqr` (x .ih x)) = (sqr` (A .ih A)))
4		dfhnorm2 8988	. 2 |- normh = {<.x, y>. | (x e. H~ /\ y = (sqr` (x .ih x)))}
5		fvex 3732	. 2 |- (sqr` (A .ih A)) e. V
6	3, 4, 5	fvopab4 3780	1 |- (A e. H~ -> (normh` A) = (sqr` (A .ih A)))

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  ` cfv 3182  (class class class)co 3963  sqrcsqr 6669  H~chil 8788   .ih csp 8793  normhcno 8794

This theorem is referenced by:  normge0t 8992  normgt0tOLD 8993  normgt0t 8994  norm0 8995  normsq 8999  norm-ii 9004  norm-iii 9006  bcsALT 9046

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hfi 8946

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-hnorm 8837

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