HomeHome Hilbert Space Explorer < Previous   Next >
Related theorems
Unicode version
Theorem adjt 9796
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
StepHypRef Expression
1 opreq1 3959 . . . . 5 |- (x = A -> (x .ih (T` y)) = (A .ih (T` y)))
2 fveq2 3715 . . . . . 6 |- (x = A -> ((adjh` T)` x) = ((adjh` T)` A))
32opreq1d 3966 . . . . 5 |- (x = A -> (((adjh` T)` x) .ih y) = (((adjh` T)` A) .ih y))
41, 3eqeq12d 1486 . . . 4 |- (x = A -> ((x .ih (T` y)) = (((adjh` T)` x) .ih y) <-> (A .ih (T` y)) = (((adjh` T)` A) .ih y)))
5 fveq2 3715 . . . . . 6 |- (y = B -> (T` y) = (T` B))
65opreq2d 3967 . . . . 5 |- (y = B -> (A .ih (T` y)) = (A .ih (T` B)))
7 opreq2 3960 . . . . 5 |- (y = B -> (((adjh` T)` A) .ih y) = (((adjh` T)` A) .ih B))
86, 7eqeq12d 1486 . . . 4 |- (y = B -> ((A .ih (T` y)) = (((adjh` T)` A) .ih y) <-> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
94, 8rcla42v 1876 . . 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 9753 . . . . . . 7 |- Fun adjh
11 funfvop 3794 . . . . . . 7 |- ((Fun adjh /\ T e. dom adjh) -> <.T, (adjh` T)>. e. adjh)
1210, 11mpan 694 . . . . . 6 |- (T e. dom adjh -> <.T, (adjh` T)>. e. adjh)
13 dfadj2 9752 . . . . . 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))}
1412, 13syl6eleq 1555 . . . . 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 3723 . . . . . 6 |- (adjh` T) e. V
16 feq1 3612 . . . . . . . 8 |- (z = T -> (z:H~-->H~ <-> T:H~-->H~))
17 fveq1 3714 . . . . . . . . . . 11 |- (z = T -> (z` y) = (T` y))
1817opreq2d 3967 . . . . . . . . . 10 |- (z = T -> (x .ih (z` y)) = (x .ih (T` y)))
1918eqeq1d 1480 . . . . . . . . 9 |- (z = T -> ((x .ih (z` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = ((w` x) .ih y)))
20192ralbidv 1677 . . . . . . . 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)))
2116, 203anbi13d 893 . . . . . . 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 3612 . . . . . . . 8 |- (w = (adjh` T) -> (w:H~-->H~ <-> (adjh` T):H~-->H~))
23 fveq1 3714 . . . . . . . . . . 11 |- (w = (adjh` T) -> (w` x) = ((adjh` T)` x))
2423opreq1d 3966 . . . . . . . . . 10 |- (w = (adjh` T) -> ((w` x) .ih y) = (((adjh` T)` x) .ih y))
2524eqeq2d 1483 . . . . . . . . 9 |- (w = (adjh` T) -> ((x .ih (T` y)) = ((w` x) .ih y) <-> (x .ih (T` y)) = (((adjh` T)` x) .ih y)))
26252ralbidv 1677 . . . . . . . 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)))
2722, 263anbi23d 894 . . . . . . 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))))
2821, 27opelopabg 2812 . . . . . 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))))
2915, 28mpan2 695 . . . . 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))))
3014, 29mpbid 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)))
31303simp3d 795 . . 3 |- (T e. dom adjh -> A.x e. H~ A.y e. H~ (x .ih (T` y)) = (((adjh` T)` x) .ih y))
329, 31syl5com 52 . 2 |- (T e. dom adjh -> ((A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B)))
33323impib 830 1 |- ((T e. dom adjh /\ A e. H~ /\ B e. H~) -> (A .ih (T` B)) = (((adjh` T)` A) .ih B))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807  <.cop 2407  {copab 2661  dom cdm 3165  Fun wfun 3171  -->wf 3173  ` cfv 3177  (class class class)co 3954  H~chil 8727   .ih csp 8732  adjhcado 8763
This theorem is referenced by:  adj2t 9797  adjadjt 9799  hmopadj2t 9804
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605  ax-hfvadd 8809  ax-hvcom 8810  ax-hvass 8811  ax-hv0cl 8812  ax-hvaddid 8813  ax-hfvmul 8814  ax-hvmulid 8815  ax-hvdistr2 8818  ax-hvmul0 8819  ax-hfi 8885  ax-his1 8888  ax-his2 8889  ax-his3 8890  ax-his4 8891
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-re 6690  df-im 6691  df-cj 6692  df-hvsub 8779  df-adjh 9715
Copyright terms: Public domain