Theorem unopt 9755

Description: Basic inner product property of a unitary operator.

Assertion

Ref	Expression
unopt	|- ((T e. UniOp /\ A e. H~ /\ B e. H~) -> ((T` A) .ih (T` B)) = (A .ih B))

Proof of Theorem unopt

Step	Hyp	Ref	Expression
1		elunopt 9716	. . . 4 |- (T e. UniOp <-> (T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y)))
2	1	pm3.27bi 326	. . 3 |- (T e. UniOp -> A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y))
3	2	3ad2ant1 798	. 2 |- ((T e. UniOp /\ A e. H~ /\ B e. H~) -> A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y))
4		fveq2 3709	. . . . . 6 |- (x = A -> (T` x) = (T` A))
5	4	opreq1d 3960	. . . . 5 |- (x = A -> ((T` x) .ih (T` y)) = ((T` A) .ih (T` y)))
6		opreq1 3953	. . . . 5 |- (x = A -> (x .ih y) = (A .ih y))
7	5, 6	eqeq12d 1481	. . . 4 |- (x = A -> (((T` x) .ih (T` y)) = (x .ih y) <-> ((T` A) .ih (T` y)) = (A .ih y)))
8		fveq2 3709	. . . . . 6 |- (y = B -> (T` y) = (T` B))
9	8	opreq2d 3961	. . . . 5 |- (y = B -> ((T` A) .ih (T` y)) = ((T` A) .ih (T` B)))
10		opreq2 3954	. . . . 5 |- (y = B -> (A .ih y) = (A .ih B))
11	9, 10	eqeq12d 1481	. . . 4 |- (y = B -> (((T` A) .ih (T` y)) = (A .ih y) <-> ((T` A) .ih (T` B)) = (A .ih B)))
12	7, 11	rcla42v 1871	. . 3 |- ((A e. H~ /\ B e. H~) -> (A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y) -> ((T` A) .ih (T` B)) = (A .ih B)))
13	12	3adant1 795	. 2 |- ((T e. UniOp /\ A e. H~ /\ B e. H~) -> (A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y) -> ((T` A) .ih (T` B)) = (A .ih B)))
14	3, 13	mpd 26	1 |- ((T e. UniOp /\ A e. H~ /\ B e. H~) -> ((T` A) .ih (T` B)) = (A .ih B))

Syntax hints:   -> wi 3   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  -onto->wfo 3170  ` cfv 3172  (class class class)co 3948  H~chil 8727   .ih csp 8732  UniOpcuo 8757

This theorem is referenced by:  unopf1ot 9756  unopnormt 9757  cnvunopt 9758  unopadjt 9759  counopt 9761

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-3an 775  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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