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Theorem nmooval 30782
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 8911 . . 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 30781 . . . . 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 2748 . . . . . . . . 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 3179 . . . . . . 7 (𝑡 = 𝑇 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))))
1615abbidv 2808 . . . . . 6 (𝑡 = 𝑇 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))})
1716supeq1d 9486 . . . . 5 (𝑡 = 𝑇 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ))
18 eqid 2737 . . . . 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 13183 . . . . . 6 < Or ℝ*
2019supex 9503 . . . . 5 sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑧)))}, ℝ*, < ) ∈ V
2117, 18, 20fvmpt 7016 . . . 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 2797 . . 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 1110 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 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  supcsup 9480  1c1 11156  *cxr 11294   < clt 11295  cle 11296  NrmCVeccnv 30603  BaseSetcba 30605  normCVcnmcv 30609   normOpOLD cnmoo 30760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-nmoo 30764
This theorem is referenced by:  nmoxr  30785  nmooge0  30786  nmorepnf  30787  nmoolb  30790  nmoubi  30791  nmoo0  30810  nmlno0lem  30812
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