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Theorem nmfnval 29580
Description: Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nmfnval (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem nmfnval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 xrltso 12522 . . 3 < Or ℝ*
21supex 8915 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 28703 . 2 ℋ ∈ V
4 cnex 10606 . 2 ℂ ∈ V
5 fveq1 6662 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
65fveq2d 6667 . . . . . . 7 (𝑡 = 𝑇 → (abs‘(𝑡𝑦)) = (abs‘(𝑇𝑦)))
76eqeq2d 2829 . . . . . 6 (𝑡 = 𝑇 → (𝑥 = (abs‘(𝑡𝑦)) ↔ 𝑥 = (abs‘(𝑇𝑦))))
87anbi2d 628 . . . . 5 (𝑡 = 𝑇 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
98rexbidv 3294 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
109abbidv 2882 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
1110supeq1d 8898 . 2 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
12 df-nmfn 29549 . 2 normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8434 1 (𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  {cab 2796  wrex 3136   class class class wbr 5057  wf 6344  cfv 6348  supcsup 8892  cc 10523  1c1 10526  *cxr 10662   < clt 10663  cle 10664  abscabs 14581  chba 28623  normcno 28627  normfncnmf 28655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-nmfn 29549
This theorem is referenced by:  nmfnxr  29583  nmfnrepnf  29584  nmfnlb  29628  nmfnleub  29629  nmfn0  29691  nmcfnexi  29755  branmfn  29809
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