Theorem ocval 28139

Description: Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ocval	|- ( H C_ ~H -> ( _|_ ` H ) = { x e. ~H | A. y e. H ( x .ih y ) = 0 } )

Distinct variable group:   x, y, H

Proof of Theorem ocval

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27856	. . 3 |- ~H e. _V
2	1	elpw2 4828	. 2 |- ( H e. ~P ~H <-> H C_ ~H )
3		raleq 3138	. . . 4 |- ( z = H -> ( A. y e. z ( x .ih y ) = 0 <-> A. y e. H ( x .ih y ) = 0 ) )
4	3	rabbidv 3189	. . 3 |- ( z = H -> { x e. ~H | A. y e. z ( x .ih y ) = 0 } = { x e. ~H | A. y e. H ( x .ih y ) = 0 } )
5		df-oc 28109	. . 3 |- _|_ = ( z e. ~P ~H |-> { x e. ~H | A. y e. z ( x .ih y ) = 0 } )
6	1	rabex 4813	. . 3 |- { x e. ~H | A. y e. H ( x .ih y ) = 0 } e. _V
7	4, 5, 6	fvmpt 6282	. 2 |- ( H e. ~P ~H -> ( _|_ ` H ) = { x e. ~H | A. y e. H ( x .ih y ) = 0 } )
8	2, 7	sylbir 225	1 |- ( H C_ ~H -> ( _|_ ` H ) = { x e. ~H | A. y e. H ( x .ih y ) = 0 } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   ~Pcpw 4158   ` cfv 5888  (class class class)co 6650   0cc0 9936   ~Hchil 27776   .ih csp 27779   _|_cort 27787

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-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-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-pw 4160  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-fv 5896  df-oc 28109

This theorem is referenced by:  ocel  28140  ocsh  28142  occon  28146  chocvali  28158