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Theorem adj1 29317
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 29270 . . . . . . 7 Fun adj
2 funfvop 6555 . . . . . . 7 ((Fun adj𝑇 ∈ dom adj) → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
31, 2mpan 682 . . . . . 6 (𝑇 ∈ dom adj → ⟨𝑇, (adj𝑇)⟩ ∈ adj)
4 dfadj2 29269 . . . . . 6 adj = {⟨𝑧, 𝑤⟩ ∣ (𝑧: ℋ⟶ ℋ ∧ 𝑤: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦))}
53, 4syl6eleq 2888 . . . . 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 6424 . . . . . 6 (adj𝑇) ∈ V
7 feq1 6237 . . . . . . . 8 (𝑧 = 𝑇 → (𝑧: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
8 fveq1 6410 . . . . . . . . . . 11 (𝑧 = 𝑇 → (𝑧𝑦) = (𝑇𝑦))
98oveq2d 6894 . . . . . . . . . 10 (𝑧 = 𝑇 → (𝑥 ·ih (𝑧𝑦)) = (𝑥 ·ih (𝑇𝑦)))
109eqeq1d 2801 . . . . . . . . 9 (𝑧 = 𝑇 → ((𝑥 ·ih (𝑧𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦)))
11102ralbidv 3170 . . . . . . . 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 1563 . . . . . . 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 6237 . . . . . . . 8 (𝑤 = (adj𝑇) → (𝑤: ℋ⟶ ℋ ↔ (adj𝑇): ℋ⟶ ℋ))
14 fveq1 6410 . . . . . . . . . . 11 (𝑤 = (adj𝑇) → (𝑤𝑥) = ((adj𝑇)‘𝑥))
1514oveq1d 6893 . . . . . . . . . 10 (𝑤 = (adj𝑇) → ((𝑤𝑥) ·ih 𝑦) = (((adj𝑇)‘𝑥) ·ih 𝑦))
1615eqeq2d 2809 . . . . . . . . 9 (𝑤 = (adj𝑇) → ((𝑥 ·ih (𝑇𝑦)) = ((𝑤𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
17162ralbidv 3170 . . . . . . . 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 1564 . . . . . . 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 5189 . . . . . 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 683 . . . . 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 224 . . . 4 (𝑇 ∈ dom adj → (𝑇: ℋ⟶ ℋ ∧ (adj𝑇): ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦)))
2221simp3d 1175 . . 3 (𝑇 ∈ dom adj → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦))
23 oveq1 6885 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝑦)))
24 fveq2 6411 . . . . . 6 (𝑥 = 𝐴 → ((adj𝑇)‘𝑥) = ((adj𝑇)‘𝐴))
2524oveq1d 6893 . . . . 5 (𝑥 = 𝐴 → (((adj𝑇)‘𝑥) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝑦))
2623, 25eqeq12d 2814 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝑥) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦)))
27 fveq2 6411 . . . . . 6 (𝑦 = 𝐵 → (𝑇𝑦) = (𝑇𝐵))
2827oveq2d 6894 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·ih (𝑇𝑦)) = (𝐴 ·ih (𝑇𝐵)))
29 oveq2 6886 . . . . 5 (𝑦 = 𝐵 → (((adj𝑇)‘𝐴) ·ih 𝑦) = (((adj𝑇)‘𝐴) ·ih 𝐵))
3028, 29eqeq12d 2814 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·ih (𝑇𝑦)) = (((adj𝑇)‘𝐴) ·ih 𝑦) ↔ (𝐴 ·ih (𝑇𝐵)) = (((adj𝑇)‘𝐴) ·ih 𝐵)))
3126, 30rspc2v 3510 . . 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 1145 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  cop 4374  {copab 4905  dom cdm 5312  Fun wfun 6095  wf 6097  cfv 6101  (class class class)co 6878  chba 28301   ·ih csp 28304  adjcado 28337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-hfvadd 28382  ax-hvcom 28383  ax-hvass 28384  ax-hv0cl 28385  ax-hvaddid 28386  ax-hfvmul 28387  ax-hvmulid 28388  ax-hvdistr2 28391  ax-hvmul0 28392  ax-hfi 28461  ax-his1 28464  ax-his2 28465  ax-his3 28466  ax-his4 28467
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-2 11376  df-cj 14180  df-re 14181  df-im 14182  df-hvsub 28353  df-adjh 29233
This theorem is referenced by:  adj2  29318  adjadj  29320  hmopadj2  29325
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