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 Description: Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
adjmo ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))
Distinct variable group:   𝑥,𝑦,𝑢,𝑇

Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 r19.26-2 3047 . . . . . 6 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) ↔ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
2 eqtr2 2630 . . . . . . . 8 (((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
32ralimi 2936 . . . . . . 7 (∀𝑦 ∈ ℋ ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
43ralimi 2936 . . . . . 6 (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
51, 4sylbir 224 . . . . 5 ((∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
6 hoeq1 28073 . . . . . 6 ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
76biimpa 500 . . . . 5 (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦)) → 𝑢 = 𝑣)
85, 7sylan2 490 . . . 4 (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
98an4s 865 . . 3 (((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
109gen2 1714 . 2 𝑢𝑣(((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
11 feq1 5939 . . . 4 (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
12 fveq1 6102 . . . . . . 7 (𝑢 = 𝑣 → (𝑢𝑥) = (𝑣𝑥))
1312oveq1d 6564 . . . . . 6 (𝑢 = 𝑣 → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
1413eqeq2d 2620 . . . . 5 (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
15142ralbidv 2972 . . . 4 (𝑢 = 𝑣 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
1611, 15anbi12d 743 . . 3 (𝑢 = 𝑣 → ((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦))))
1716mo4 2505 . 2 (∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ↔ ∀𝑢𝑣(((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦))) → 𝑢 = 𝑣))
1810, 17mpbir 220 1 ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃*wmo 2459  ∀wral 2896  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ℋchil 27160   ·ih csp 27163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-hfvadd 27241  ax-hvcom 27242  ax-hvass 27243  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvdistr2 27250  ax-hvmul0 27251  ax-hfi 27320  ax-his2 27324  ax-his3 27325  ax-his4 27326 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-sub 10147  df-neg 10148  df-hvsub 27212 This theorem is referenced by:  funadj  28129  adjeu  28132  cnlnadjeui  28320
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