HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmopval Structured version   Visualization version   GIF version

Theorem nmopval 31876
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 13184 . . 3 < Or ℝ*
21supex 9504 . 2 sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ) ∈ V
3 ax-hilex 31019 . 2 ℋ ∈ V
4 fveq1 6904 . . . . . . . 8 (𝑡 = 𝑇 → (𝑡𝑦) = (𝑇𝑦))
54fveq2d 6909 . . . . . . 7 (𝑡 = 𝑇 → (norm‘(𝑡𝑦)) = (norm‘(𝑇𝑦)))
65eqeq2d 2747 . . . . . 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 3178 . . . 4 (𝑡 = 𝑇 → (∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦))) ↔ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))))
98abbidv 2807 . . 3 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦)))} = {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))})
109supeq1d 9487 . 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 31859 . 2 normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑦)))}, ℝ*, < ))
122, 3, 3, 10, 11fvmptmap 8922 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 395   = wceq 1539  {cab 2713  wrex 3069   class class class wbr 5142  wf 6556  cfv 6560  supcsup 9481  1c1 11157  *cxr 11295   < clt 11296  cle 11297  chba 30939  normcno 30943  normopcnop 30965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-hilex 31019
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-nmop 31859
This theorem is referenced by:  nmopxr  31886  nmoprepnf  31887  nmoplb  31927  nmopub  31928  nmopnegi  31985  nmop0  32006  nmlnop0iALT  32015  nmopun  32034  nmcopexi  32047  pjnmopi  32168
  Copyright terms: Public domain W3C validator