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Theorem adjmo 31861
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

Proof of Theorem adjmo
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 r19.26-2 3136 . . . . . 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 2759 . . . . . . 7 (((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
322ralimi 3121 . . . . . 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))
41, 3sylbir 235 . . . . 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))
5 hoeq1 31859 . . . . . 6 ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
65biimpa 476 . . . . 5 (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦)) → 𝑢 = 𝑣)
74, 6sylan2 593 . . . 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)
87an4s 660 . . 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)
98gen2 1793 . 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)
10 feq1 6717 . . . 4 (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11 fveq1 6906 . . . . . . 7 (𝑢 = 𝑣 → (𝑢𝑥) = (𝑣𝑥))
1211oveq1d 7446 . . . . . 6 (𝑢 = 𝑣 → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
1312eqeq2d 2746 . . . . 5 (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
14132ralbidv 3219 . . . 4 (𝑢 = 𝑣 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
1510, 14anbi12d 632 . . 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))))
1615mo4 2564 . 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))
179, 16mpbir 231 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 395  wal 1535   = wceq 1537  ∃*wmo 2536  wral 3059  wf 6559  cfv 6563  (class class class)co 7431  chba 30948   ·ih csp 30951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-hfvadd 31029  ax-hvcom 31030  ax-hvass 31031  ax-hv0cl 31032  ax-hvaddid 31033  ax-hfvmul 31034  ax-hvmulid 31035  ax-hvdistr2 31038  ax-hvmul0 31039  ax-hfi 31108  ax-his2 31112  ax-his3 31113  ax-his4 31114
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-ltxr 11298  df-sub 11492  df-neg 11493  df-hvsub 31000
This theorem is referenced by:  funadj  31915  adjeu  31918  cnlnadjeui  32106
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