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Theorem adj1 29702
Description: Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adj1 ((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))

Proof of Theorem adj1
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funadj 29655 . . . . . . 7 Fun adj
2 funfvop 6813 . . . . . . 7 ((Fun adj𝑇 ∈ dom adj) → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
31, 2mpan 688 . . . . . 6 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
4 dfadj2 29654 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))}
53, 4eleqtrdi 2921 . . . . 5 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))})
6 fvex 6676 . . . . . 6 (adj𝑇) ∈ V
7 feq1 6488 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8 fveq1 6662 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑦) = (𝑇𝑦))
98oveq2d 7164 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑥 ·ih (𝑧𝑦)) = (𝑥 ·ih (𝑇𝑦)))
109eqeq1d 2821 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
11102ralbidv 3197 . . . . . . . 8 (𝑧 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
127, 113anbi13d 1431 . . . . . . 7 (𝑧 = 𝑇 → ((𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦))))
13 feq1 6488 . . . . . . . 8 (𝑤 = (adj𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj𝑇): ℋ⟶ ℋ))
14 fveq1 6662 . . . . . . . . . . 11 (𝑤 = (adj𝑇) → (𝑤𝑥) = ((adj𝑇)‘𝑥))
1514oveq1d 7163 . . . . . . . . . 10 (𝑤 = (adj𝑇) → ((𝑤𝑥) ·ih 𝑦) = (((adj𝑇)‘𝑥) ·ih 𝑦))
1615eqeq2d 2830 . . . . . . . . 9 (𝑤 = (adj𝑇) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
17162ralbidv 3197 . . . . . . . 8 (𝑤 = (adj𝑇) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
1813, 173anbi23d 1432 . . . . . . 7 (𝑤 = (adj𝑇) → ((𝑇: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
1912, 18opelopabg 5416 . . . . . 6 ((𝑇 ∈ dom adj ∧ (adj𝑇) ∈ V) → (⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
206, 19mpan2 689 . . . . 5 (𝑇 ∈ dom adj → (⟨𝑇, (adj𝑇)⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))} ↔ (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))))
215, 20mpbid 234 . . . 4 (𝑇 ∈ dom adj → (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
2221simp3d 1138 . . 3 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))
23 oveq1 7155 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
24 fveq2 6663 . . . . . 6 (𝑥 = 𝐴 → ((adj𝑇)‘𝑥) = ((adj𝑇)‘𝐴))
2524oveq1d 7163 . . . . 5 (𝑥 = 𝐴 → (((adj𝑇)‘𝑥) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝑦))
2623, 25eqeq12d 2835 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦)))
27 fveq2 6663 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
2827oveq2d 7164 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
29 oveq2 7156 . . . . 5 (𝑦 = 𝐵 → (((adj𝑇)‘𝐴) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝐵))
3028, 29eqeq12d 2835 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3126, 30rspc2v 3631 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3222, 31syl5com 31 . 2 (𝑇 ∈ dom adj → ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
33323impib 1110 1 ((𝑇 ∈ dom adj𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493  cop 4565  {copab 5119  dom cdm 5548  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7148  chba 28688   ·ih csp 28691  adjcado 28724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-hfvadd 28769  ax-hvcom 28770  ax-hvass 28771  ax-hv0cl 28772  ax-hvaddid 28773  ax-hfvmul 28774  ax-hvmulid 28775  ax-hvdistr2 28778  ax-hvmul0 28779  ax-hfi 28848  ax-his1 28851  ax-his2 28852  ax-his3 28853  ax-his4 28854
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-2 11692  df-cj 14450  df-re 14451  df-im 14452  df-hvsub 28740  df-adjh 29618
This theorem is referenced by:  adj2  29703  adjadj  29705  hmopadj2  29710
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