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Theorem nmfnval 29635
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 12511 . . 3 < Or ℝ*
21supex 8903 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 28758 . 2 ℋ ∈ V
4 cnex 10594 . 2 ℂ ∈ V
5 fveq1 6643 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
65fveq2d 6648 . . . . . . 7 (𝑡 = 𝑇 → (abs‘(𝑡𝑦)) = (abs‘(𝑇𝑦)))
76eqeq2d 2831 . . . . . 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 3284 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))))
109abbidv 2884 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))})
1110supeq1d 8886 . 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 29604 . 2 normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑦)))}, ℝ*, < ))
132, 3, 4, 11, 12fvmptmap 8421 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 398   = wceq 1537  {cab 2798  wrex 3126   class class class wbr 5040  wf 6325  cfv 6329  supcsup 8880  cc 10511  1c1 10514  *cxr 10650   < clt 10651  cle 10652  abscabs 14571  chba 28678  normcno 28682  normfncnmf 28710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-cnex 10569  ax-resscn 10570  ax-pre-lttri 10587  ax-pre-lttrn 10588  ax-hilex 28758
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-po 5448  df-so 5449  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-ov 7134  df-oprab 7135  df-mpo 7136  df-er 8265  df-map 8384  df-en 8486  df-dom 8487  df-sdom 8488  df-sup 8882  df-pnf 10653  df-mnf 10654  df-xr 10655  df-ltxr 10656  df-nmfn 29604
This theorem is referenced by:  nmfnxr  29638  nmfnrepnf  29639  nmfnlb  29683  nmfnleub  29684  nmfn0  29746  nmcfnexi  29810  branmfn  29864
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