MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmooval Structured version   Visualization version   GIF version

Theorem nmooval 27746
Description: The operator norm function. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSet‘𝑈)
nmoofval.2 𝑌 = (BaseSet‘𝑊)
nmoofval.3 𝐿 = (normCV𝑈)
nmoofval.4 𝑀 = (normCV𝑊)
nmoofval.6 𝑁 = (𝑈 normOpOLD 𝑊)
Assertion
Ref Expression
nmooval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
Distinct variable groups:   𝑥,𝑧,𝑈   𝑥,𝑊,𝑧   𝑧,𝑋   𝑥,𝑌   𝑥,𝑇,𝑧
Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmooval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 nmoofval.2 . . . . 5 𝑌 = (BaseSet‘𝑊)
2 fvex 6239 . . . . 5 (BaseSet‘𝑊) ∈ V
31, 2eqeltri 2726 . . . 4 𝑌 ∈ V
4 nmoofval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
5 fvex 6239 . . . . 5 (BaseSet‘𝑈) ∈ V
64, 5eqeltri 2726 . . . 4 𝑋 ∈ V
73, 6elmap 7928 . . 3 (𝑇 ∈ (𝑌𝑚 𝑋) ↔ 𝑇:𝑋𝑌)
8 nmoofval.3 . . . . . 6 𝐿 = (normCV𝑈)
9 nmoofval.4 . . . . . 6 𝑀 = (normCV𝑊)
10 nmoofval.6 . . . . . 6 𝑁 = (𝑈 normOpOLD 𝑊)
114, 1, 8, 9, 10nmoofval 27745 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
1211fveq1d 6231 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑇) = ((𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))‘𝑇))
13 fveq1 6228 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑡𝑧) = (𝑇𝑧))
1413fveq2d 6233 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑀‘(𝑡𝑧)) = (𝑀‘(𝑇𝑧)))
1514eqeq2d 2661 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑇𝑧))))
1615anbi2d 740 . . . . . . . 8 (𝑡 = 𝑇 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1716rexbidv 3081 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1817abbidv 2770 . . . . . 6 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))})
1918supeq1d 8393 . . . . 5 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
20 eqid 2651 . . . . 5 (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))
21 xrltso 12012 . . . . . 6 < Or ℝ*
2221supex 8410 . . . . 5 sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ) ∈ V
2319, 20, 22fvmpt 6321 . . . 4 (𝑇 ∈ (𝑌𝑚 𝑋) → ((𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))‘𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
2412, 23sylan9eq 2705 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ (𝑌𝑚 𝑋)) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
257, 24sylan2br 492 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
26253impa 1278 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  supcsup 8387  1c1 9975  *cxr 10111   < clt 10112  cle 10113  NrmCVeccnv 27567  BaseSetcba 27569  normCVcnmcv 27573   normOpOLD cnmoo 27724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-nmoo 27728
This theorem is referenced by:  nmoxr  27749  nmooge0  27750  nmorepnf  27751  nmoolb  27754  nmoubi  27755  nmoo0  27774  nmlno0lem  27776
  Copyright terms: Public domain W3C validator