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Theorem nmfnval 31905
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 13180 . . 3 < Or ℝ*
21supex 9501 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 31028 . 2 ℋ ∈ V
4 cnex 11234 . 2 ℂ ∈ V
5 fveq1 6906 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
65fveq2d 6911 . . . . . . 7 (𝑡 = 𝑇 → (abs‘(𝑡𝑦)) = (abs‘(𝑇𝑦)))
76eqeq2d 2746 . . . . . 6 (𝑡 = 𝑇 → (𝑥 = (abs‘(𝑡𝑦)) ↔ 𝑥 = (abs‘(𝑇𝑦))))
87anbi2d 630 . . . . 5 (𝑡 = 𝑇 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
98rexbidv 3177 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
109abbidv 2806 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
1110supeq1d 9484 . 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 31874 . 2 normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8920 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 395   = wceq 1537  {cab 2712  wrex 3068   class class class wbr 5148  wf 6559  cfv 6563  supcsup 9478  cc 11151  1c1 11154  *cxr 11292   < clt 11293  cle 11294  abscabs 15270  chba 30948  normcno 30952  normfncnmf 30980
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-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-hilex 31028
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-rmo 3378  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-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-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-nmfn 31874
This theorem is referenced by:  nmfnxr  31908  nmfnrepnf  31909  nmfnlb  31953  nmfnleub  31954  nmfn0  32016  nmcfnexi  32080  branmfn  32134
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