Theorem elunop 28731

Description: Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
elunop	|- ( T e. UniOp <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )

Distinct variable group:   x, y, T

Proof of Theorem elunop

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( T e. UniOp -> T e. _V )
2		fof 6115	. . . 4 |- ( T : ~H -onto-> ~H -> T : ~H --> ~H )
3		ax-hilex 27856	. . . 4 |- ~H e. _V
4		fex 6490	. . . 4 |- ( ( T : ~H --> ~H /\ ~H e. _V ) -> T e. _V )
5	2, 3, 4	sylancl 694	. . 3 |- ( T : ~H -onto-> ~H -> T e. _V )
6	5	adantr 481	. 2 |- ( ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) -> T e. _V )
7		foeq1 6111	. . . 4 |- ( t = T -> ( t : ~H -onto-> ~H <-> T : ~H -onto-> ~H ) )
8		fveq1 6190	. . . . . . 7 |- ( t = T -> ( t ` x ) = ( T ` x ) )
9		fveq1 6190	. . . . . . 7 |- ( t = T -> ( t ` y ) = ( T ` y ) )
10	8, 9	oveq12d 6668	. . . . . 6 |- ( t = T -> ( ( t ` x ) .ih ( t ` y ) ) = ( ( T ` x ) .ih ( T ` y ) ) )
11	10	eqeq1d 2624	. . . . 5 |- ( t = T -> ( ( ( t ` x ) .ih ( t ` y ) ) = ( x .ih y ) <-> ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )
12	11	2ralbidv 2989	. . . 4 |- ( t = T -> ( A. x e. ~H A. y e. ~H ( ( t ` x ) .ih ( t ` y ) ) = ( x .ih y ) <-> A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )
13	7, 12	anbi12d 747	. . 3 |- ( t = T -> ( ( t : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( t ` x ) .ih ( t ` y ) ) = ( x .ih y ) ) <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) ) )
14		df-unop 28702	. . 3 |- UniOp = { t | ( t : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( t ` x ) .ih ( t ` y ) ) = ( x .ih y ) ) }
15	13, 14	elab2g 3353	. 2 |- ( T e. _V -> ( T e. UniOp <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) ) )
16	1, 6, 15	pm5.21nii 368	1 |- ( T e. UniOp <-> ( T : ~H -onto-> ~H /\ A. x e. ~H A. y e. ~H ( ( T ` x ) .ih ( T ` y ) ) = ( x .ih y ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   ~Hchil 27776   .ih csp 27779   UniOpcuo 27806

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-rep 4771  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-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-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-unop 28702

This theorem is referenced by:  unop  28774  unopf1o  28775  cnvunop  28777  counop  28780  idunop  28837  lnopunii  28871  elunop2  28872