Theorem hmop 11483

Description: Basic inner product property of a Hermitian operator.

Assertion

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

Proof of Theorem hmop

Step	Hyp	Ref	Expression
1		elhmop 11437	. . . 4 |- (T e. HrmOp <-> (T:~H-->~H /\ A.x e. ~H A.y e. ~H (x .ih (T` y)) = ((T` x) .ih y)))
2	1	simprbi 353	. . 3 |- (T e. HrmOp -> A.x e. ~H A.y e. ~H (x .ih (T` y)) = ((T` x) .ih y))
3	2	3ad2ant1 897	. 2 |- ((T e. HrmOp /\ A e. ~H /\ B e. ~H) -> A.x e. ~H A.y e. ~H (x .ih (T` y)) = ((T` x) .ih y))
4		opreq1 4889	. . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
5		fveq2 4681	. . . . . 6 |- (x = A -> (T` x) = (T` A))
6	5	opreq1d 4897	. . . . 5 |- (x = A -> ((T` x) .ih y) = ((T` A) .ih y))
7	4, 6	eqeq12d 1899	. . . 4 |- (x = A -> ((x .ih (T` y)) = ((T` x) .ih y) <-> (A .ih (T` y)) = ((T` A) .ih y)))
8		fveq2 4681	. . . . . 6 |- (y = B -> (T` y) = (T` B))
9	8	opreq2d 4898	. . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
10		opreq2 4890	. . . . 5 |- (y = B -> ((T` A) .ih y) = ((T` A) .ih B))
11	9, 10	eqeq12d 1899	. . . 4 |- (y = B -> ((A .ih (T` y)) = ((T` A) .ih y) <-> (A .ih (T` B)) = ((T` A) .ih B)))
12	7, 11	rcla42v 2384	. . 3 |- ((A e. ~H /\ B e. ~H) -> (A.x e. ~H A.y e. ~H (x .ih (T` y)) = ((T` x) .ih y) -> (A .ih (T` B)) = ((T` A) .ih B)))
13	12	3adant1 894	. 2 |- ((T e. HrmOp /\ A e. ~H /\ B e. ~H) -> (A.x e. ~H A.y e. ~H (x .ih (T` y)) = ((T` x) .ih y) -> (A .ih (T` B)) = ((T` A) .ih B)))
14	3, 13	mpd 29	1 |- ((T e. HrmOp /\ A e. ~H /\ B e. ~H) -> (A .ih (T` B)) = ((T` A) .ih B))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  -->wf 3994  ` cfv 3998  (class class class)co 4884  ~Hchil 10420   .ih csp 10425  HrmOpcho 10451

This theorem is referenced by:  hmopre 11484  hmopadj 11500  hmoplin 11503  eighmre 11524  eighmorth 11525  hmopbdopiHIL 11549  hmops 11582  hmopm 11583  hmopco 11585  leopsq 11700  hmopidmchlem 11722  hmopidmpji 11724

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-3an 860  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-fv 4014  df-opr 4886  df-hmop 11407

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