Theorem hmopt 9841

Description: Basic inner product property of a Hermitian operator.

Assertion

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

Proof of Theorem hmopt

Step	Hyp	Ref	Expression
1		elhmopt 9795	. . . 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	pm3.27bi 326	. . 3 |- (T e. HrmOp -> A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((T` x) .ih y))
3	2	3ad2ant1 802	. 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 3974	. . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
5		fveq2 3730	. . . . . 6 |- (x = A -> (T` x) = (T` A))
6	5	opreq1d 3981	. . . . 5 |- (x = A -> ((T` x) .ih y) = ((T` A) .ih y))
7	4, 6	eqeq12d 1492	. . . 4 |- (x = A -> ((x .ih (T` y)) = ((T` x) .ih y) <-> (A .ih (T` y)) = ((T` A) .ih y)))
8		fveq2 3730	. . . . . 6 |- (y = B -> (T` y) = (T` B))
9	8	opreq2d 3982	. . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
10		opreq2 3975	. . . . 5 |- (y = B -> ((T` A) .ih y) = ((T` A) .ih B))
11	9, 10	eqeq12d 1492	. . . 4 |- (y = B -> ((A .ih (T` y)) = ((T` A) .ih y) <-> (A .ih (T` B)) = ((T` A) .ih B)))
12	7, 11	rcla42v 1883	. . 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 799	. 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 26	1 |- ((T e. HrmOp /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = ((T` A) .ih B))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  -->wf 3184  ` cfv 3188  (class class class)co 3969  H~chil 8783   .ih csp 8788  HrmOpcho 8814

This theorem is referenced by:  hmopret 9842  hmopadjt 9858  hmoplint 9861  eighmret 9882  eighmortht 9883  hmopbdopHIL 9907  hmopst 9940  hmopmt 9941  hmopcot 9943  leopsqt 10057  hmopidmchlem 10073  hmopidmpj 10075

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-3an 779  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-fv 3204  df-opr 3971  df-hmop 9765

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