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Theorem nmfnval 31895
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 13183 . . 3 < Or ℝ*
21supex 9503 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 31018 . 2 ℋ ∈ V
4 cnex 11236 . 2 ℂ ∈ V
5 fveq1 6905 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
65fveq2d 6910 . . . . . . 7 (𝑡 = 𝑇 → (abs‘(𝑡𝑦)) = (abs‘(𝑇𝑦)))
76eqeq2d 2748 . . . . . 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 3179 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
109abbidv 2808 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
1110supeq1d 9486 . 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 31864 . 2 normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8921 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 1540  {cab 2714  wrex 3070   class class class wbr 5143  wf 6557  cfv 6561  supcsup 9480  cc 11153  1c1 11156  *cxr 11294   < clt 11295  cle 11296  abscabs 15273  chba 30938  normcno 30942  normfncnmf 30970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-nmfn 31864
This theorem is referenced by:  nmfnxr  31898  nmfnrepnf  31899  nmfnlb  31943  nmfnleub  31944  nmfn0  32006  nmcfnexi  32070  branmfn  32124
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