Theorem adjeqt 9859

Description: A property that determines the adjoint of a Hilbert space operator.

Assertion

Ref	Expression
adjeqt	|- ((T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) -> (adjh` T) = S)

Distinct variable groups:   x,y,S   x,T,y

Proof of Theorem adjeqt

Step	Hyp	Ref	Expression
1		funadj 9813	. . 3 |- Fun adjh
2		funopfvg 3752	. . 3 |- ((S e. V /\ Fun adjh) -> (<.T, S>. e. adjh -> (adjh` T) = S))
3	1, 2	mpan2 696	. 2 |- (S e. V -> (<.T, S>. e. adjh -> (adjh` T) = S))
4		ax-hilex 8869	. . . 4 |- H~ e. V
5		fex 3652	. . . 4 |- ((S:H~-->H~ /\ H~ e. V) -> S e. V)
6	4, 5	mpan2 696	. . 3 |- (S:H~-->H~ -> S e. V)
7	6	3ad2ant2 801	. 2 |- ((T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) -> S e. V)
8		feq1 3620	. . . . . . . . 9 |- (z = T -> (z:H~-->H~ <-> T:H~-->H~))
9		fveq1 3723	. . . . . . . . . . . 12 |- (z = T -> (z` x) = (T` x))
10	9	opreq1d 3975	. . . . . . . . . . 11 |- (z = T -> ((z` x) .ih y) = ((T` x) .ih y))
11	10	eqeq1d 1483	. . . . . . . . . 10 |- (z = T -> (((z` x) .ih y) = (x .ih (w` y)) <-> ((T` x) .ih y) = (x .ih (w` y))))
12	11	2ralbidv 1680	. . . . . . . . 9 |- (z = T -> (A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y)) <-> A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (w` y))))
13	8, 12	3anbi13d 895	. . . . . . . 8 |- (z = T -> ((z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y))) <-> (T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (w` y)))))
14		feq1 3620	. . . . . . . . 9 |- (w = S -> (w:H~-->H~ <-> S:H~-->H~))
15		fveq1 3723	. . . . . . . . . . . 12 |- (w = S -> (w` y) = (S` y))
16	15	opreq2d 3976	. . . . . . . . . . 11 |- (w = S -> (x .ih (w` y)) = (x .ih (S` y)))
17	16	eqeq2d 1486	. . . . . . . . . 10 |- (w = S -> (((T` x) .ih y) = (x .ih (w` y)) <-> ((T` x) .ih y) = (x .ih (S` y))))
18	17	2ralbidv 1680	. . . . . . . . 9 |- (w = S -> (A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (w` y)) <-> A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))))
19	14, 18	3anbi23d 896	. . . . . . . 8 |- (w = S -> ((T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (w` y))) <-> (T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)))))
20	13, 19	opelopabg 2817	. . . . . . 7 |- ((T e. V /\ S e. V) -> (<.T, S>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y)))} <-> (T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)))))
21		fex 3652	. . . . . . . 8 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
22	4, 21	mpan2 696	. . . . . . 7 |- (T:H~-->H~ -> T e. V)
23	20, 22, 6	syl2an 454	. . . . . 6 |- ((T:H~-->H~ /\ S:H~-->H~) -> (<.T, S>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y)))} <-> (T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)))))
24		df-adjh 9775	. . . . . . 7 |- adjh = {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y)))}
25	24	eleq2i 1538	. . . . . 6 |- (<.T, S>. e. adjh <-> <.T, S>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ ((z` x) .ih y) = (x .ih (w` y)))})
26	23, 25	syl5bb 532	. . . . 5 |- ((T:H~-->H~ /\ S:H~-->H~) -> (<.T, S>. e. adjh <-> (T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)))))
27		df-3an 777	. . . . . 6 |- ((T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) <-> ((T:H~-->H~ /\ S:H~-->H~) /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))))
28	27	baibr 686	. . . . 5 |- ((T:H~-->H~ /\ S:H~-->H~) -> (A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)) <-> (T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y)))))
29	26, 28	bitr4d 531	. . . 4 |- ((T:H~-->H~ /\ S:H~-->H~) -> (<.T, S>. e. adjh <-> A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))))
30	29	biimpar 417	. . 3 |- (((T:H~-->H~ /\ S:H~-->H~) /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) -> <.T, S>. e. adjh)
31	30	3impa 828	. 2 |- ((T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) -> <.T, S>. e. adjh)
32	3, 7, 31	sylc 68	1 |- ((T:H~-->H~ /\ S:H~-->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih y) = (x .ih (S` y))) -> (adjh` T) = S)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  <.cop 2411  {copab 2666  Fun wfun 3176  -->wf 3178  ` cfv 3182  (class class class)co 3963  H~chil 8788   .ih csp 8793  adj_hcado 8824

This theorem is referenced by:  unopadj2t 9862  hmopadjt 9863  adj0 9919  adjmult 10025  adjaddt 10026

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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-re 6751  df-im 6752  df-cj 6753  df-hvsub 8840  df-adjh 9775

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