Theorem adjt 9773

Description: Property of an adjoint Hilbert space operator.

Assertion

Ref	Expression
adjt	|- ((T e. dom adjh /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B))

Proof of Theorem adjt

Step	Hyp	Ref	Expression
1		opreq1 3953	. . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
2		fveq2 3709	. . . . . 6 |- (x = A -> ((adjh` T)` x) = ((adjh` T)` A))
3	2	opreq1d 3960	. . . . 5 |- (x = A -> (((adjh` T)` x) .ih y) = (((adjh` T)` A) .ih y))
4	1, 3	eqeq12d 1481	. . . 4 |- (x = A -> ((x .ih (T` y)) = (((adjh` T)` x) .ih y) <-> (A .ih (T` y)) = (((adjh` T)` A) .ih y)))
5		fveq2 3709	. . . . . 6 |- (y = B -> (T` y) = (T` B))
6	5	opreq2d 3961	. . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
7		opreq2 3954	. . . . 5 |- (y = B -> (((adjh` T)` A) .ih y) = (((adjh` T)` A) .ih B))
8	6, 7	eqeq12d 1481	. . . 4 |- (y = B -> ((A .ih (T` y)) = (((adjh` T)` A) .ih y) <-> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
9	4, 8	rcla42v 1871	. . 3 |- ((A e. H~ /\ B e. H~) -> (A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
10		funadj 9730	. . . . . . 7 |- Fun adjh
11		funfvop 3788	. . . . . . 7 |- ((Fun adjh /\ T e. dom adjh) -> <.T, (adjh` T)>. e. adjh)
12	10, 11	mpan 693	. . . . . 6 |- (T e. dom adjh -> <.T, (adjh` T)>. e. adjh)
13		dfadj2 9729	. . . . . 6 |- adjh = {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))}
14	12, 13	syl6eleq 1550	. . . . 5 |- (T e. dom adjh -> <.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))})
15		fvex 3717	. . . . . 6 |- (adjh` T) e. V
16		feq1 3606	. . . . . . . 8 |- (z = T -> (z:H~-->H~ <-> T:H~-->H~))
17		fveq1 3708	. . . . . . . . . . 11 |- (z = T -> (z` y) = (T` y))
18	17	opreq2d 3961	. . . . . . . . . 10 |- (z = T -> (x .ih (z` y)) = (x .ih (T` y)))
19	18	eqeq1d 1475	. . . . . . . . 9 |- (z = T -> ((x .ih (z` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = ((w` x) .ih y)))
20	19	2ralbidv 1672	. . . . . . . 8 |- (z = T -> (A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y) <-> A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y)))
21	16, 20	3anbi13d 892	. . . . . . 7 |- (z = T -> ((z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y)) <-> (T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y))))
22		feq1 3606	. . . . . . . 8 |- (w = (adjh` T) -> (w:H~-->H~ <-> (adjh` T):H~-->H~))
23		fveq1 3708	. . . . . . . . . . 11 |- (w = (adjh` T) -> (w` x) = ((adjh` T)` x))
24	23	opreq1d 3960	. . . . . . . . . 10 |- (w = (adjh` T) -> ((w` x) .ih y) = (((adjh` T)` x) .ih y))
25	24	eqeq2d 1478	. . . . . . . . 9 |- (w = (adjh` T) -> ((x .ih (T` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
26	25	2ralbidv 1672	. . . . . . . 8 |- (w = (adjh` T) -> (A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y) <-> A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
27	22, 26	3anbi23d 893	. . . . . . 7 |- (w = (adjh` T) -> ((T:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = ((w` x) .ih y)) <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
28	21, 27	opelopabg 2806	. . . . . 6 |- ((T e. dom adjh /\ (adjh` T) e. V) -> (<.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))} <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
29	15, 28	mpan2 694	. . . . 5 |- (T e. dom adjh -> (<.T, (adjh` T)>. e. {<.z, w>. | (z:H~-->H~ /\ w:H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (z` y)) = ((w` x) .ih y))} <-> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))))
30	14, 29	mpbid 195	. . . 4 |- (T e. dom adjh -> (T:H~-->H~ /\ (adjh` T):H~-->H~ /\ A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
31	30	3simp3d 794	. . 3 |- (T e. dom adjh -> A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))
32	9, 31	syl5com 52	. 2 |- (T e. dom adjh -> ((A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
33	32	3impib 829	1 |- ((T e. dom adjh /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802  <.cop 2401  {copab 2656  dom cdm 3160  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:  adj2t 9774  adjadjt 9776  hmopadj2t 9781

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-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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