Theorem elunop 11436

Description: Property defining a unitary Hilbert space operator.

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

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (T e. UniOp -> T e. _V)
2		fof 4617	. . . 4 |- (T:~H-onto->~H -> T:~H-->~H)
3		ax-hilex 10501	. . . . 5 |- ~H e. _V
4		fex 4595	. . . . 5 |- ((T:~H-->~H /\ ~H e. _V) -> T e. _V)
5	3, 4	mpan2 760	. . . 4 |- (T:~H-->~H -> T e. _V)
6	2, 5	syl 12	. . 3 |- (T:~H-onto->~H -> T e. _V)
7	6	adantr 425	. 2 |- ((T:~H-onto->~H /\ A.x e. ~H A.y e. ~H ((T` x) .ih (T` y)) = (x .ih y)) -> T e. _V)
8		foeq1 4613	. . . 4 |- (t = T -> (t:~H-onto->~H <-> T:~H-onto->~H))
9		fveq1 4680	. . . . . . 7 |- (t = T -> (t` x) = (T` x))
10		fveq1 4680	. . . . . . 7 |- (t = T -> (t` y) = (T` y))
11	9, 10	opreq12d 4900	. . . . . 6 |- (t = T -> ((t` x) .ih (t` y)) = ((T` x) .ih (T` y)))
12	11	eqeq1d 1892	. . . . 5 |- (t = T -> (((t` x) .ih (t` y)) = (x .ih y) <-> ((T` x) .ih (T` y)) = (x .ih y)))
13	12	2ralbidv 2140	. . . 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)))
14	8, 13	anbi12d 690	. . 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))))
15		df-unop 11406	. . 3 |- UniOp = {t | (t:~H-onto->~H /\ A.x e. ~H A.y e. ~H ((t` x) .ih (t` y)) = (x .ih y))}
16	14, 15	elab2g 2406	. 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))))
17	1, 7, 16	pm5.21nii 743	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 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  ~Hchil 10420   .ih csp 10425  UniOpcuo 10450

This theorem is referenced by:  unop 11476  unopf1o 11477  cnvunop 11479  counop 11482  idunop 11539  lnopunii 11574  elunop2 11575

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-hilex 10501

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-rex 2110  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-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-unop 11406

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