Theorem elunopt 9716

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 1808	. 2 |- (T e. UniOp -> T e. V)
2		fof 3657	. . . 4 |- (T:H~-onto->H~ -> T:H~-->H~)
3		ax-hilex 8790	. . . . 5 |- H~ e. V
4		fex 3637	. . . . 5 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
5	3, 4	mpan2 694	. . . 4 |- (T:H~-->H~ -> T e. V)
6	2, 5	syl 10	. . 3 |- (T:H~-onto->H~ -> T e. V)
7	6	adantr 389	. 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 3653	. . . 4 |- (t = T -> (t:H~-onto->H~ <-> T:H~-onto->H~))
9		fveq1 3708	. . . . . . 7 |- (t = T -> (t` x) = (T` x))
10		fveq1 3708	. . . . . . 7 |- (t = T -> (t` y) = (T` y))
11	9, 10	opreq12d 3963	. . . . . 6 |- (t = T -> ((t` x) .ih (t` y)) = ((T` x) .ih (T` y)))
12	11	eqeq1d 1475	. . . . 5 |- (t = T -> (((t` x) .ih (t` y)) = (x .ih y) <-> ((T` x) .ih (T` y)) = (x .ih y)))
13	12	2ralbidv 1672	. . . 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 626	. . 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 9686	. . 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 1891	. 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 677	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 953   e. wcel 955  A.wral 1637  Vcvv 1802  -->wf 3168  -onto->wfo 3170  ` cfv 3172  (class class class)co 3948  H~chil 8727   .ih csp 8732  UniOpcuo 8757

This theorem is referenced by:  unopt 9755  unopf1ot 9756  cnvunopt 9758  counopt 9761  idunop 9818  lnopuni 9852  elunop2t 9853

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-opr 3950  df-unop 9686

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