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Theorem nmooval 28555
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)
21fvexi 6677 . . . 4 𝑌 ∈ V
3 nmoofval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
43fvexi 6677 . . . 4 𝑋 ∈ V
52, 4elmap 8433 . . 3 (𝑇 ∈ (𝑌m 𝑋) ↔ 𝑇:𝑋𝑌)
6 nmoofval.3 . . . . . 6 𝐿 = (normCV𝑈)
7 nmoofval.4 . . . . . 6 𝑀 = (normCV𝑊)
8 nmoofval.6 . . . . . 6 𝑁 = (𝑈 normOpOLD 𝑊)
93, 1, 6, 7, 8nmoofval 28554 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
109fveq1d 6665 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑇) = ((𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))‘𝑇))
11 fveq1 6662 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑡𝑧) = (𝑇𝑧))
1211fveq2d 6667 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑀‘(𝑡𝑧)) = (𝑀‘(𝑇𝑧)))
1312eqeq2d 2835 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑇𝑧))))
1413anbi2d 631 . . . . . . . 8 (𝑡 = 𝑇 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1514rexbidv 3289 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1615abbidv 2888 . . . . . 6 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))})
1716supeq1d 8909 . . . . 5 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
18 eqid 2824 . . . . 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*, < ))
19 xrltso 12533 . . . . . 6 < Or ℝ*
2019supex 8926 . . . . 5 sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ) ∈ V
2117, 18, 20fvmpt 6761 . . . 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*, < ))
2210, 21sylan9eq 2879 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ (𝑌m 𝑋)) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
235, 22sylan2br 597 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
24233impa 1107 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 399  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wrex 3134   class class class wbr 5053  cmpt 5133  wf 6341  cfv 6345  (class class class)co 7151  m cmap 8404  supcsup 8903  1c1 10538  *cxr 10674   < clt 10675  cle 10676  NrmCVeccnv 28376  BaseSetcba 28378  normCVcnmcv 28382   normOpOLD cnmoo 28533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-pre-lttri 10611  ax-pre-lttrn 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-po 5462  df-so 5463  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-sup 8905  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-nmoo 28537
This theorem is referenced by:  nmoxr  28558  nmooge0  28559  nmorepnf  28560  nmoolb  28563  nmoubi  28564  nmoo0  28583  nmlno0lem  28585
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