Theorem dfhnorm2 10621

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 10469	. 2 |- normh = {<.x, y>. | (x e. dom dom .ih /\ y = (sqr` (x .ih x)))}
2		ax-hfi 10579	. . . . . . . 8 |- .ih :(~H X. ~H)-->CC
3	2	fdmi 4568	. . . . . . 7 |- dom .ih = (~H X. ~H)
4	3	dmeqi 4158	. . . . . 6 |- dom dom .ih = dom (~H X. ~H)
5		dmxpid 4179	. . . . . 6 |- dom (~H X. ~H) = ~H
6	4, 5	eqtr2i 1909	. . . . 5 |- ~H = dom dom .ih
7	6	eleq2i 1961	. . . 4 |- (x e. ~H <-> x e. dom dom .ih )
8	7	anbi1i 539	. . 3 |- ((x e. ~H /\ y = (sqr` (x .ih x))) <-> (x e. dom dom .ih /\ y = (sqr` (x .ih x))))
9	8	opabbii 3402	. 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	eqtr4i 1911	1 |- normh = {<.x, y>. | (x e. ~H /\ y = (sqr` (x .ih x)))}

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  {copab 3395   X. cxp 3984  dom cdm 3986  ` cfv 3998  (class class class)co 4884  CCcc 6384  sqrcsqr 7919  ~Hchil 10420   .ih csp 10425  normhcno 10426

This theorem is referenced by:  normf 10622  normval 10623  hilnormi 10663

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-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-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-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-br 3339  df-opab 3396  df-xp 4000  df-dm 4004  df-fn 4009  df-f 4010  df-hnorm 10469

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