Theorem hmoval 8470

Description: The set of Hermitian (self-adjoint) operators on a normed complex vector space.

Hypotheses

Ref	Expression
hmoval.8	|- H = (HmOp` U)
hmoval.9	|- A = (UadjU)

Assertion

Ref	Expression
hmoval	|- (U e. NrmCVec -> H = {t e. dom A | (A` t) = t})

Distinct variable groups:   t,A   t,U

Proof of Theorem hmoval

Step	Hyp	Ref	Expression
1		opreq12 3970	. . . . . . . 8 |- ((u = U /\ u = U) -> (uadju) = (UadjU))
2	1	anidms 434	. . . . . . 7 |- (u = U -> (uadju) = (UadjU))
3		hmoval.9	. . . . . . 7 |- A = (UadjU)
4	2, 3	syl6eqr 1525	. . . . . 6 |- (u = U -> (uadju) = A)
5	4	dmeqd 3313	. . . . 5 |- (u = U -> dom ( uadju) = dom A)
6		rabeq 1809	. . . . 5 |- (dom ( uadju) = dom A -> {t e. dom ( uadju) | ((uadju)` t) = t} = {t e. dom A | ((uadju)` t) = t})
7	5, 6	syl 10	. . . 4 |- (u = U -> {t e. dom ( uadju) | ((uadju)` t) = t} = {t e. dom A | ((uadju)` t) = t})
8	4	fveq1d 3726	. . . . . 6 |- (u = U -> ((uadju)` t) = (A` t))
9	8	eqeq1d 1483	. . . . 5 |- (u = U -> (((uadju)` t) = t <-> (A` t) = t))
10	9	rabbisdv 1807	. . . 4 |- (u = U -> {t e. dom A | ((uadju)` t) = t} = {t e. dom A | (A` t) = t})
11	7, 10	eqtrd 1507	. . 3 |- (u = U -> {t e. dom ( uadju) | ((uadju)` t) = t} = {t e. dom A | (A` t) = t})
12		df-hmo 8412	. . 3 |- HmOp = {<.u, o>. | (u e. NrmCVec /\ o = {t e. dom ( uadju) | ((uadju)` t) = t})}
13		oprex 3983	. . . . . 6 |- (UadjU) e. V
14	3, 13	eqeltr 1544	. . . . 5 |- A e. V
15	14	dmex 3360	. . . 4 |- dom A e. V
16	15	rabex 2725	. . 3 |- {t e. dom A | (A` t) = t} e. V
17	11, 12, 16	fvopab4 3780	. 2 |- (U e. NrmCVec -> (HmOp` U) = {t e. dom A | (A` t) = t})
18		hmoval.8	. 2 |- H = (HmOp` U)
19	17, 18	syl5eq 1519	1 |- (U e. NrmCVec -> H = {t e. dom A | (A` t) = t})

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811  dom cdm 3170  ` cfv 3182  (class class class)co 3963  NrmCVeccnv 8203  adjcaj 8409  HmOpchmo 8410

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-9 965  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-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rex 1650  df-rab 1652  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-fv 3198  df-opr 3965  df-hmo 8412

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