Theorem normvalt 8985

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 3976	. . . 4 |- ((x = A /\ x = A) -> (x .ih x) = (A .ih A))
2	1	anidms 436	. . 3 |- (x = A -> (x .ih x) = (A .ih A))
3	2	fveq2d 3734	. 2 |- (x = A -> (sqr` (x .ih x)) = (sqr` (A .ih A)))
4		dfhnorm2 8983	. 2 |- normh = {<.x, y>. | (x e. H~ /\ y = (sqr` (x .ih x)))}
5		fvex 3738	. 2 |- (sqr` (A .ih A)) e. V
6	3, 4, 5	fvopab4 3786	1 |- (A e. H~ -> (normh` A) = (sqr` (A .ih A)))

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  ` cfv 3188  (class class class)co 3969  sqrcsqr 6670  H~chil 8783   .ih csp 8788  normhcno 8789

This theorem is referenced by:  normge0t 8987  normgt0tOLD 8988  normgt0t 8989  norm0 8990  normsq 8994  norm-ii 8999  norm-iii 9001  bcsALT 9041

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hfi 8941

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-hnorm 8832

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