Theorem hmopt 9846

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 9800	. . . 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 800	. 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 3968	. . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
5		fveq2 3724	. . . . . 6 |- (x = A -> (T` x) = (T` A))
6	5	opreq1d 3975	. . . . 5 |- (x = A -> ((T` x) .ih y) = ((T` A) .ih y))
7	4, 6	eqeq12d 1489	. . . 4 |- (x = A -> ((x .ih (T` y)) = ((T` x) .ih y) <-> (A .ih (T` y)) = ((T` A) .ih y)))
8		fveq2 3724	. . . . . 6 |- (y = B -> (T` y) = (T` B))
9	8	opreq2d 3976	. . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
10		opreq2 3969	. . . . 5 |- (y = B -> ((T` A) .ih y) = ((T` A) .ih B))
11	9, 10	eqeq12d 1489	. . . 4 |- (y = B -> ((A .ih (T` y)) = ((T` A) .ih y) <-> (A .ih (T` B)) = ((T` A) .ih B)))
12	7, 11	rcla42v 1880	. . 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 797	. 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 775   = wceq 956   e. wcel 958  A.wral 1645  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8788   .ih csp 8793  HrmOpcho 8819

This theorem is referenced by:  hmopret 9847  hmopadjt 9863  hmoplint 9866  eighmret 9887  eighmortht 9888  hmopbdopHIL 9912  hmopst 9945  hmopmt 9946  hmopcot 9948  leopsqt 10062  hmopidmchlem 10078  hmopidmpj 10080

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-hmop 9770

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