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Theorem nmoofval 27001
 Description: The operator norm function. (Contributed by NM, 6-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
nmoofval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
Distinct variable groups:   𝑥,𝑡,𝑧,𝑈   𝑡,𝑊,𝑥,𝑧   𝑡,𝑋,𝑧   𝑡,𝑌,𝑥   𝑡,𝐿   𝑡,𝑀
Allowed substitution hints:   𝐿(𝑥,𝑧)   𝑀(𝑥,𝑧)   𝑁(𝑥,𝑧,𝑡)   𝑋(𝑥)   𝑌(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (𝑈 normOpOLD 𝑊)
2 fveq2 6103 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2662 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54oveq2d 6565 . . . 4 (𝑢 = 𝑈 → ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) = ((BaseSet‘𝑤) ↑𝑚 𝑋))
6 fveq2 6103 . . . . . . . . . . 11 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCV𝑈)
86, 7syl6eqr 2662 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = 𝐿)
98fveq1d 6105 . . . . . . . . 9 (𝑢 = 𝑈 → ((normCV𝑢)‘𝑧) = (𝐿𝑧))
109breq1d 4593 . . . . . . . 8 (𝑢 = 𝑈 → (((normCV𝑢)‘𝑧) ≤ 1 ↔ (𝐿𝑧) ≤ 1))
1110anbi1d 737 . . . . . . 7 (𝑢 = 𝑈 → ((((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
124, 11rexeqbidv 3130 . . . . . 6 (𝑢 = 𝑈 → (∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))))
1312abbidv 2728 . . . . 5 (𝑢 = 𝑈 → {𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))})
1413supeq1d 8235 . . . 4 (𝑢 = 𝑈 → sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ))
155, 14mpteq12dv 4663 . . 3 (𝑢 = 𝑈 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
16 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 nmoofval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
1816, 17syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918oveq1d 6564 . . . 4 (𝑤 = 𝑊 → ((BaseSet‘𝑤) ↑𝑚 𝑋) = (𝑌𝑚 𝑋))
20 fveq2 6103 . . . . . . . . . . 11 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCV𝑊)
2220, 21syl6eqr 2662 . . . . . . . . . 10 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
2322fveq1d 6105 . . . . . . . . 9 (𝑤 = 𝑊 → ((normCV𝑤)‘(𝑡𝑧)) = (𝑀‘(𝑡𝑧)))
2423eqeq2d 2620 . . . . . . . 8 (𝑤 = 𝑊 → (𝑥 = ((normCV𝑤)‘(𝑡𝑧)) ↔ 𝑥 = (𝑀‘(𝑡𝑧))))
2524anbi2d 736 . . . . . . 7 (𝑤 = 𝑊 → (((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2625rexbidv 3034 . . . . . 6 (𝑤 = 𝑊 → (∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧))) ↔ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))))
2726abbidv 2728 . . . . 5 (𝑤 = 𝑊 → {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))} = {𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))})
2827supeq1d 8235 . . . 4 (𝑤 = 𝑊 → sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < ) = sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < ))
2919, 28mpteq12dv 4663 . . 3 (𝑤 = 𝑊 → (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
30 df-nmoo 26984 . . 3 normOpOLD = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ (𝑡 ∈ ((BaseSet‘𝑤) ↑𝑚 (BaseSet‘𝑢)) ↦ sup({𝑥 ∣ ∃𝑧 ∈ (BaseSet‘𝑢)(((normCV𝑢)‘𝑧) ≤ 1 ∧ 𝑥 = ((normCV𝑤)‘(𝑡𝑧)))}, ℝ*, < )))
31 ovex 6577 . . . 4 (𝑌𝑚 𝑋) ∈ V
3231mptex 6390 . . 3 (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpt2 6694 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 normOpOLD 𝑊) = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
341, 33syl5eq 2656 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑁 = (𝑡 ∈ (𝑌𝑚 𝑋) ↦ sup({𝑥 ∣ ∃𝑧𝑋 ((𝐿𝑧) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑡𝑧)))}, ℝ*, < )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  supcsup 8229  1c1 9816  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  NrmCVeccnv 26823  BaseSetcba 26825  normCVcnmcv 26829   normOpOLD cnmoo 26980 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-sup 8231  df-nmoo 26984 This theorem is referenced by:  nmooval  27002  hhnmoi  28144
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