Theorem dfhnorm2 8983

Description: Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96.

Assertion

Ref	Expression
dfhnorm2	|- normh = {<.x, y>. | (x e. H~ /\ y = (sqr` (x .ih x)))}

Distinct variable group:   x,y

Proof of Theorem dfhnorm2

Step	Hyp	Ref	Expression
1		df-hnorm 8832	. 2 |- normh = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}
2		ax-hfi 8941	. . . . . . . 8 |- .ih :(H~ X. H~)-->CC
3	2	fdmi 3638	. . . . . . 7 |- dom .ih = (H~ X. H~)
4	3	dmeqi 3318	. . . . . 6 |- dom dom .ih = dom (H~ X. H~)
5		dmxpid 3339	. . . . . 6 |- dom (H~ X. H~) = H~
6	4, 5	eqtr2 1499	. . . . 5 |- H~ = dom dom .ih
7	6	eleq2i 1541	. . . 4 |- (x e. H~ <-> x e. dom dom .ih )
8	7	anbi1i 483	. . 3 |- ((x e. H~ /\ y = (sqr` (x .ih x))) <-> (x e. dom dom .ih /\ y = (sqr` (x .ih x))))
9	8	opabbii 2676	. 2 |- {<.x, y>. | (x e. H~ /\ y = (sqr` (x .ih x)))} = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}
10	1, 9	eqtr4 1501	1 |- normh = {<.x, y>. | (x e. H~ /\ y = (sqr` (x .ih x)))}

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  {copab 2671   X. cxp 3174  dom cdm 3176  ` cfv 3188  (class class class)co 3969  CCcc 5244  sqrcsqr 6670  H~chil 8783   .ih csp 8788  normhcno 8789

This theorem is referenced by:  normf 8984  normvalt 8985  hilnorm 9025

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-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-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-br 2625  df-opab 2672  df-xp 3190  df-dm 3194  df-fn 3199  df-f 3200  df-hnorm 8832

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