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Theorem adjmo 29235
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 3275 . . . . . 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 2847 . . . . . . 7 (((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)) → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
322ralimi 3162 . . . . . 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 227 . . . . 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 29233 . . . . . 6 ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
65biimpa 470 . . . . 5 (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦)) → 𝑢 = 𝑣)
74, 6sylan2 586 . . . 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 650 . . 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 1895 . 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 6259 . . . 4 (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11 fveq1 6432 . . . . . . 7 (𝑢 = 𝑣 → (𝑢𝑥) = (𝑣𝑥))
1211oveq1d 6920 . . . . . 6 (𝑢 = 𝑣 → ((𝑢𝑥) ·ih 𝑦) = ((𝑣𝑥) ·ih 𝑦))
1312eqeq2d 2835 . . . . 5 (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇𝑦)) = ((𝑣𝑥) ·ih 𝑦)))
14132ralbidv 3198 . . . 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 624 . . 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 2638 . 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 223 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 386  wal 1654   = wceq 1656  ∃*wmo 2603  wral 3117  wf 6119  cfv 6123  (class class class)co 6905  chba 28320   ·ih csp 28323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-hfvadd 28401  ax-hvcom 28402  ax-hvass 28403  ax-hv0cl 28404  ax-hvaddid 28405  ax-hfvmul 28406  ax-hvmulid 28407  ax-hvdistr2 28410  ax-hvmul0 28411  ax-hfi 28480  ax-his2 28484  ax-his3 28485  ax-his4 28486
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-ltxr 10396  df-sub 10587  df-neg 10588  df-hvsub 28372
This theorem is referenced by:  funadj  29289  adjeu  29292  cnlnadjeui  29480
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