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Theorem adjmo 28561
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 3059 . . . . . 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 2641 . . . . . . 7 (((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
322ralimi 2948 . . . . . 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 225 . . . . 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 28559 . . . . . 6 ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
65biimpa 501 . . . . 5 (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦)) → 𝑢 = 𝑣)
74, 6sylan2 491 . . . 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 868 . . 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 1720 . 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 5988 . . . 4 (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11 fveq1 6152 . . . . . . 7 (𝑢 = 𝑣 → (𝑢𝑥) = (𝑣𝑥))
1211oveq1d 6625 . . . . . 6 (𝑢 = 𝑣 → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
1312eqeq2d 2631 . . . . 5 (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
14132ralbidv 2984 . . . 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 746 . . 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 2516 . 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 221 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 384  wal 1478   = wceq 1480  ∃*wmo 2470  wral 2907  wf 5848  cfv 5852  (class class class)co 6610  chil 27646   ·ih csp 27649
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-hfvadd 27727  ax-hvcom 27728  ax-hvass 27729  ax-hv0cl 27730  ax-hvaddid 27731  ax-hfvmul 27732  ax-hvmulid 27733  ax-hvdistr2 27736  ax-hvmul0 27737  ax-hfi 27806  ax-his2 27810  ax-his3 27811  ax-his4 27812
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-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-ltxr 10031  df-sub 10220  df-neg 10221  df-hvsub 27698
This theorem is referenced by:  funadj  28615  adjeu  28618  cnlnadjeui  28806
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