Theorem elunopt 9794

Description: Property defining a unitary Hilbert space operator.

Assertion

Ref	Expression
elunopt	|- (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 elunopt

Step	Hyp	Ref	Expression
1		elisset 1820	. 2 |- (T e. UniOp -> T e. V)
2		fof 3678	. . . 4 |- (T:H~-onto->H~ -> T:H~-->H~)
3		ax-hilex 8864	. . . . 5 |- H~ e. V
4		fex 3658	. . . . 5 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
5	3, 4	mpan2 698	. . . 4 |- (T:H~-->H~ -> T e. V)
6	2, 5	syl 10	. . 3 |- (T:H~-onto->H~ -> T e. V)
7	6	adantr 391	. 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 3674	. . . 4 |- (t = T -> (t:H~-onto->H~ <-> T:H~-onto->H~))
9		fveq1 3729	. . . . . . 7 |- (t = T -> (t` x) = (T` x))
10		fveq1 3729	. . . . . . 7 |- (t = T -> (t` y) = (T` y))
11	9, 10	opreq12d 3984	. . . . . 6 |- (t = T -> ((t` x) .ih (t` y)) = ((T` x) .ih (T` y)))
12	11	eqeq1d 1486	. . . . 5 |- (t = T -> (((t` x) .ih (t` y)) = (x .ih y) <-> ((T` x) .ih (T` y)) = (x .ih y)))
13	12	2ralbidv 1683	. . . 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 630	. . 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 9764	. . 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 1903	. 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 681	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 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  -->wf 3184  -onto->wfo 3186  ` cfv 3188  (class class class)co 3969  H~chil 8783   .ih csp 8788  UniOpcuo 8813

This theorem is referenced by:  unopt 9834  unopf1ot 9835  cnvunopt 9837  counopt 9840  idunop 9897  lnopuni 9932  elunop2t 9933

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-unop 9764

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