Theorem hfmmval 28598

Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Assertion

Ref	Expression
hfmmval	|- ( ( A e. CC /\ T : ~H --> CC ) -> ( A .fn T ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )

Distinct variable groups:   x, A   x, T

Proof of Theorem hfmmval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 10017	. . 3 |- CC e. _V
2		ax-hilex 27856	. . 3 |- ~H e. _V
3	1, 2	elmap 7886	. 2 |- ( T e. ( CC ^m ~H ) <-> T : ~H --> CC )
4		oveq1 6657	. . . 4 |- ( f = A -> ( f x. ( g ` x ) ) = ( A x. ( g ` x ) ) )
5	4	mpteq2dv 4745	. . 3 |- ( f = A -> ( x e. ~H |-> ( f x. ( g ` x ) ) ) = ( x e. ~H |-> ( A x. ( g ` x ) ) ) )
6		fveq1 6190	. . . . 5 |- ( g = T -> ( g ` x ) = ( T ` x ) )
7	6	oveq2d 6666	. . . 4 |- ( g = T -> ( A x. ( g ` x ) ) = ( A x. ( T ` x ) ) )
8	7	mpteq2dv 4745	. . 3 |- ( g = T -> ( x e. ~H |-> ( A x. ( g ` x ) ) ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )
9		df-hfmul 28593	. . 3 |- .fn = ( f e. CC , g e. ( CC ^m ~H ) |-> ( x e. ~H |-> ( f x. ( g ` x ) ) ) )
10	2	mptex 6486	. . 3 |- ( x e. ~H |-> ( A x. ( T ` x ) ) ) e. _V
11	5, 8, 9, 10	ovmpt2 6796	. 2 |- ( ( A e. CC /\ T e. ( CC ^m ~H ) ) -> ( A .fn T ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )
12	3, 11	sylan2br 493	1 |- ( ( A e. CC /\ T : ~H --> CC ) -> ( A .fn T ) = ( x e. ~H |-> ( A x. ( T ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   x. cmul 9941   ~Hchil 27776   .fn chft 27799

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-hilex 27856

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-hfmul 28593

This theorem is referenced by:  hfmval  28603  brafnmul  28810  kbass2  28976