Theorem normval 27981

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. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

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

Proof of Theorem normval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq12 6659	. . . 4 |- ( ( x = A /\ x = A ) -> ( x .ih x ) = ( A .ih A ) )
2	1	anidms 677	. . 3 |- ( x = A -> ( x .ih x ) = ( A .ih A ) )
3	2	fveq2d 6195	. 2 |- ( x = A -> ( sqr ` ( x .ih x ) ) = ( sqr ` ( A .ih A ) ) )
4		dfhnorm2 27979	. 2 |- normh = ( x e. ~H |-> ( sqr ` ( x .ih x ) ) )
5		fvex 6201	. 2 |- ( sqr ` ( A .ih A ) ) e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( A e. ~H -> ( normh ` A ) = ( sqr ` ( A .ih A ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   sqrcsqrt 13973   ~Hchil 27776   .ih csp 27779   normhcno 27780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hfi 27936

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-hnorm 27825

This theorem is referenced by:  normge0  27983  normgt0  27984  norm0  27985  normsqi  27989  norm-ii-i  27994  norm-iii-i  27996  bcsiALT  28036