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Theorem adj1 28638
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 28591 . . . . . . 7 Fun adj
2 funfvop 6285 . . . . . . 7 ((Fun adj𝑇 ∈ dom adj) → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
31, 2mpan 705 . . . . . 6 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
4 dfadj2 28590 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))}
53, 4syl6eleq 2708 . . . . 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 6158 . . . . . 6 (adj𝑇) ∈ V
7 feq1 5983 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8 fveq1 6147 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑦) = (𝑇𝑦))
98oveq2d 6620 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑥 ·ih (𝑧𝑦)) = (𝑥 ·ih (𝑇𝑦)))
109eqeq1d 2623 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
11102ralbidv 2983 . . . . . . . 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 1398 . . . . . . 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 5983 . . . . . . . 8 (𝑤 = (adj𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj𝑇): ℋ⟶ ℋ))
14 fveq1 6147 . . . . . . . . . . 11 (𝑤 = (adj𝑇) → (𝑤𝑥) = ((adj𝑇)‘𝑥))
1514oveq1d 6619 . . . . . . . . . 10 (𝑤 = (adj𝑇) → ((𝑤𝑥) ·ih 𝑦) = (((adj𝑇)‘𝑥) ·ih 𝑦))
1615eqeq2d 2631 . . . . . . . . 9 (𝑤 = (adj𝑇) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
17162ralbidv 2983 . . . . . . . 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 1399 . . . . . . 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 4953 . . . . . 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 706 . . . . 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 222 . . . 4 (𝑇 ∈ dom adj → (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
2221simp3d 1073 . . 3 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))
23 oveq1 6611 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
24 fveq2 6148 . . . . . 6 (𝑥 = 𝐴 → ((adj𝑇)‘𝑥) = ((adj𝑇)‘𝐴))
2524oveq1d 6619 . . . . 5 (𝑥 = 𝐴 → (((adj𝑇)‘𝑥) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝑦))
2623, 25eqeq12d 2636 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦)))
27 fveq2 6148 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
2827oveq2d 6620 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
29 oveq2 6612 . . . . 5 (𝑦 = 𝐵 → (((adj𝑇)‘𝐴) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝐵))
3028, 29eqeq12d 2636 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3126, 30rspc2v 3306 . . 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 1259 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cop 4154  {copab 4672  dom cdm 5074  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  chil 27622   ·ih csp 27625  adjcado 27658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-hfvadd 27703  ax-hvcom 27704  ax-hvass 27705  ax-hv0cl 27706  ax-hvaddid 27707  ax-hfvmul 27708  ax-hvmulid 27709  ax-hvdistr2 27712  ax-hvmul0 27713  ax-hfi 27782  ax-his1 27785  ax-his2 27786  ax-his3 27787  ax-his4 27788
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-2 11023  df-cj 13773  df-re 13774  df-im 13775  df-hvsub 27674  df-adjh 28554
This theorem is referenced by:  adj2  28639  adjadj  28641  hmopadj2  28646
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