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Theorem nmopval 30840
Description: Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
nmopval (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
Distinct variable group:   𝑥,𝑦,𝑇

Proof of Theorem nmopval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 xrltso 13066 . . 3 < Or ℝ*
21supex 9404 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 29983 . 2 ℋ ∈ V
4 fveq1 6842 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
54fveq2d 6847 . . . . . . 7 (𝑡 = 𝑇 → (norm‘(𝑡𝑦)) = (norm‘(𝑇𝑦)))
65eqeq2d 2744 . . . . . 6 (𝑡 = 𝑇 → (𝑥 = (norm‘(𝑡𝑦)) ↔ 𝑥 = (norm‘(𝑇𝑦))))
76anbi2d 630 . . . . 5 (𝑡 = 𝑇 → (((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦))) ↔ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))))
87rexbidv 3172 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))))
98abbidv 2802 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
109supeq1d 9387 . 2 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
11 df-nmop 30823 . 2 normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦)))}, ℝ*, < ))
122, 3, 3, 10, 11fvmptmap 8822 1 (𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  {cab 2710  wrex 3070   class class class wbr 5106  wf 6493  cfv 6497  supcsup 9381  1c1 11057  *cxr 11193   < clt 11194  cle 11195  chba 29903  normcno 29907  normopcnop 29929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-hilex 29983
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-nmop 30823
This theorem is referenced by:  nmopxr  30850  nmoprepnf  30851  nmoplb  30891  nmopub  30892  nmopnegi  30949  nmop0  30970  nmlnop0iALT  30979  nmopun  30998  nmcopexi  31011  pjnmopi  31132
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