Theorem adjeqt 9775

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 9730	. . 3 |- Fun adjh
2		funopfvg 3737	. . 3 |- ((S e. V /\ Fun adjh) -> (<.T, S>. e. adjh -> (adjh` T) = S))
3	1, 2	mpan2 694	. 2 |- (S e. V -> (<.T, S>. e. adjh -> (adjh` T) = S))
4		ax-hilex 8790	. . . 4 |- H~ e. V
5		fex 3637	. . . 4 |- ((S:H~-->H~ /\ H~ e. V) -> S e. V)
6	4, 5	mpan2 694	. . 3 |- (S:H~-->H~ -> S e. V)
7	6	3ad2ant2 799	. 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 3606	. . . . . . . . 9 |- (z = T -> (z:H~-->H~ <-> T:H~-->H~))
9		fveq1 3708	. . . . . . . . . . . 12 |- (z = T -> (z` x) = (T` x))
10	9	opreq1d 3960	. . . . . . . . . . 11 |- (z = T -> ((z` x) .ih y) = ((T` x) .ih y))
11	10	eqeq1d 1475	. . . . . . . . . 10 |- (z = T -> (((z` x) .ih y) = (x .ih (w` y)) <-> ((T` x) .ih y) = (x .ih (w` y))))
12	11	2ralbidv 1672	. . . . . . . . 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 892	. . . . . . . 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 3606	. . . . . . . . 9 |- (w = S -> (w:H~-->H~ <-> S:H~-->H~))
15		fveq1 3708	. . . . . . . . . . . 12 |- (w = S -> (w` y) = (S` y))
16	15	opreq2d 3961	. . . . . . . . . . 11 |- (w = S -> (x .ih (w` y)) = (x .ih (S` y)))
17	16	eqeq2d 1478	. . . . . . . . . 10 |- (w = S -> (((T` x) .ih y) = (x .ih (w` y)) <-> ((T` x) .ih y) = (x .ih (S` y))))
18	17	2ralbidv 1672	. . . . . . . . 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 893	. . . . . . . 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 2806	. . . . . . 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 3637	. . . . . . . 8 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
22	4, 21	mpan2 694	. . . . . . 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 9692	. . . . . . 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 1530	. . . . . 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 530	. . . . 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 775	. . . . . 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 684	. . . . 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 529	. . . 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 826	. 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 773   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802  <.cop 2401  {copab 2656  Fun wfun 3166  -->wf 3168  ` cfv 3172  (class class class)co 3948  H~chil 8727   .ih csp 8732  adj_hcado 8763

This theorem is referenced by:  unopadj2t 9778  hmopadjt 9779  adj0 9834  adjmult 9939  adjaddt 9940

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790  ax-hfvadd 8791  ax-hvcom 8792  ax-hvass 8793  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvdistr2 8800  ax-hvmul0 8801  ax-hfi 8867  ax-his1 8870  ax-his2 8871  ax-his3 8872  ax-his4 8873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-re 6682  df-im 6683  df-cj 6684  df-hvsub 8779  df-adjh 9692

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