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Theorem adj1 31952
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 31905 . . . . . . 7 Fun adj
2 funfvop 7070 . . . . . . 7 ((Fun adj𝑇 ∈ dom adj) → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
31, 2mpan 690 . . . . . 6 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
4 dfadj2 31904 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))}
53, 4eleqtrdi 2851 . . . . 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 6919 . . . . . 6 (adj𝑇) ∈ V
7 feq1 6716 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8 fveq1 6905 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑦) = (𝑇𝑦))
98oveq2d 7447 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑥 ·ih (𝑧𝑦)) = (𝑥 ·ih (𝑇𝑦)))
109eqeq1d 2739 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
11102ralbidv 3221 . . . . . . . 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 1440 . . . . . . 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 6716 . . . . . . . 8 (𝑤 = (adj𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj𝑇): ℋ⟶ ℋ))
14 fveq1 6905 . . . . . . . . . . 11 (𝑤 = (adj𝑇) → (𝑤𝑥) = ((adj𝑇)‘𝑥))
1514oveq1d 7446 . . . . . . . . . 10 (𝑤 = (adj𝑇) → ((𝑤𝑥) ·ih 𝑦) = (((adj𝑇)‘𝑥) ·ih 𝑦))
1615eqeq2d 2748 . . . . . . . . 9 (𝑤 = (adj𝑇) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
17162ralbidv 3221 . . . . . . . 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 1441 . . . . . . 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 5543 . . . . . 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 691 . . . . 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 232 . . . 4 (𝑇 ∈ dom adj → (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
2221simp3d 1145 . . 3 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))
23 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
24 fveq2 6906 . . . . . 6 (𝑥 = 𝐴 → ((adj𝑇)‘𝑥) = ((adj𝑇)‘𝐴))
2524oveq1d 7446 . . . . 5 (𝑥 = 𝐴 → (((adj𝑇)‘𝑥) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝑦))
2623, 25eqeq12d 2753 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦)))
27 fveq2 6906 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
2827oveq2d 7447 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
29 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (((adj𝑇)‘𝐴) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝐵))
3028, 29eqeq12d 2753 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3126, 30rspc2v 3633 . . 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 1117 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  cop 4632  {copab 5205  dom cdm 5685  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  chba 30938   ·ih csp 30941  adjcado 30974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-hfvadd 31019  ax-hvcom 31020  ax-hvass 31021  ax-hv0cl 31022  ax-hvaddid 31023  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvdistr2 31028  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his2 31102  ax-his3 31103  ax-his4 31104
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-2 12329  df-cj 15138  df-re 15139  df-im 15140  df-hvsub 30990  df-adjh 31868
This theorem is referenced by:  adj2  31953  adjadj  31955  hmopadj2  31960
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