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Theorem nmooval 30791
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 6920 . . . 4 𝑌 ∈ V
3 nmoofval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
43fvexi 6920 . . . 4 𝑋 ∈ V
52, 4elmap 8909 . . 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 30790 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
109fveq1d 6908 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑁𝑇) = ((𝑡 ∈ (𝑌m 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))‘𝑇))
11 fveq1 6905 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑡𝑧) = (𝑇𝑧))
1211fveq2d 6910 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑀‘(𝑡𝑧)) = (𝑀‘(𝑇𝑧)))
1312eqeq2d 2745 . . . . . . . . 9 (𝑡 = 𝑇 → (𝑥 = (𝑀‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑇𝑧))))
1413anbi2d 630 . . . . . . . 8 (𝑡 = 𝑇 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1514rexbidv 3176 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1615abbidv 2805 . . . . . 6 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))})
1716supeq1d 9483 . . . . 5 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
18 eqid 2734 . . . . 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 13179 . . . . . 6 < Or ℝ*
2019supex 9500 . . . . 5 sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ) ∈ V
2117, 18, 20fvmpt 7015 . . . 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 2794 . . 3 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ (𝑌m 𝑋)) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
235, 22sylan2br 595 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
24233impa 1109 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 395  w3a 1086   = wceq 1536  wcel 2105  {cab 2711  wrex 3067   class class class wbr 5147  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  supcsup 9477  1c1 11153  *cxr 11291   < clt 11292  cle 11293  NrmCVeccnv 30612  BaseSetcba 30614  normCVcnmcv 30618   normOpOLD cnmoo 30769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-nmoo 30773
This theorem is referenced by:  nmoxr  30794  nmooge0  30795  nmorepnf  30796  nmoolb  30799  nmoubi  30800  nmoo0  30819  nmlno0lem  30821
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