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Theorem nmoolb 28598
 Description: A lower bound for an operator norm. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoolb.1 𝑋 = (BaseSet‘𝑈)
nmoolb.2 𝑌 = (BaseSet‘𝑊)
nmoolb.l 𝐿 = (normCV𝑈)
nmoolb.m 𝑀 = (normCV𝑊)
nmoolb.3 𝑁 = (𝑈 normOpOLD 𝑊)
Assertion
Ref Expression
nmoolb (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑀‘(𝑇𝐴)) ≤ (𝑁𝑇))

Proof of Theorem nmoolb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoolb.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
2 nmoolb.m . . . . . 6 𝑀 = (normCV𝑊)
31, 2nmosetre 28591 . . . . 5 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))} ⊆ ℝ)
4 ressxr 10692 . . . . 5 ℝ ⊆ ℝ*
53, 4sstrdi 3929 . . . 4 ((𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))} ⊆ ℝ*)
653adant1 1127 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))} ⊆ ℝ*)
7 fveq2 6655 . . . . . . . 8 (𝑦 = 𝐴 → (𝐿𝑦) = (𝐿𝐴))
87breq1d 5044 . . . . . . 7 (𝑦 = 𝐴 → ((𝐿𝑦) ≤ 1 ↔ (𝐿𝐴) ≤ 1))
9 2fveq3 6660 . . . . . . . 8 (𝑦 = 𝐴 → (𝑀‘(𝑇𝑦)) = (𝑀‘(𝑇𝐴)))
109eqeq2d 2809 . . . . . . 7 (𝑦 = 𝐴 → ((𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦)) ↔ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝐴))))
118, 10anbi12d 633 . . . . . 6 (𝑦 = 𝐴 → (((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦))) ↔ ((𝐿𝐴) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝐴)))))
12 eqid 2798 . . . . . . 7 (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝐴))
1312biantru 533 . . . . . 6 ((𝐿𝐴) ≤ 1 ↔ ((𝐿𝐴) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝐴))))
1411, 13bitr4di 292 . . . . 5 (𝑦 = 𝐴 → (((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦))) ↔ (𝐿𝐴) ≤ 1))
1514rspcev 3572 . . . 4 ((𝐴𝑋 ∧ (𝐿𝐴) ≤ 1) → ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦))))
16 fvex 6668 . . . . 5 (𝑀‘(𝑇𝐴)) ∈ V
17 eqeq1 2802 . . . . . . 7 (𝑥 = (𝑀‘(𝑇𝐴)) → (𝑥 = (𝑀‘(𝑇𝑦)) ↔ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦))))
1817anbi2d 631 . . . . . 6 (𝑥 = (𝑀‘(𝑇𝐴)) → (((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦))) ↔ ((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦)))))
1918rexbidv 3257 . . . . 5 (𝑥 = (𝑀‘(𝑇𝐴)) → (∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦))) ↔ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦)))))
2016, 19elab 3616 . . . 4 ((𝑀‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))} ↔ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ (𝑀‘(𝑇𝐴)) = (𝑀‘(𝑇𝑦))))
2115, 20sylibr 237 . . 3 ((𝐴𝑋 ∧ (𝐿𝐴) ≤ 1) → (𝑀‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))})
22 supxrub 12725 . . 3 (({𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))} ⊆ ℝ* ∧ (𝑀‘(𝑇𝐴)) ∈ {𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))}) → (𝑀‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))}, ℝ*, < ))
236, 21, 22syl2an 598 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑀‘(𝑇𝐴)) ≤ sup({𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))}, ℝ*, < ))
24 nmoolb.1 . . . 4 𝑋 = (BaseSet‘𝑈)
25 nmoolb.l . . . 4 𝐿 = (normCV𝑈)
26 nmoolb.3 . . . 4 𝑁 = (𝑈 normOpOLD 𝑊)
2724, 1, 25, 2, 26nmooval 28590 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))}, ℝ*, < ))
2827adantr 484 . 2 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑁𝑇) = sup({𝑥 ∣ ∃𝑦𝑋 ((𝐿𝑦) ≤ 1 ∧ 𝑥 = (𝑀‘(𝑇𝑦)))}, ℝ*, < ))
2923, 28breqtrrd 5062 1 (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) ∧ (𝐴𝑋 ∧ (𝐿𝐴) ≤ 1)) → (𝑀‘(𝑇𝐴)) ≤ (𝑁𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107   ⊆ wss 3883   class class class wbr 5034  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  supcsup 8906  ℝcr 10543  1c1 10545  ℝ*cxr 10681   < clt 10682   ≤ cle 10683  NrmCVeccnv 28411  BaseSetcba 28413  normCVcnmcv 28417   normOpOLD cnmoo 28568 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-er 8290  df-map 8409  df-en 8511  df-dom 8512  df-sdom 8513  df-sup 8908  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-vc 28386  df-nv 28419  df-va 28422  df-ba 28423  df-sm 28424  df-0v 28425  df-nmcv 28427  df-nmoo 28572 This theorem is referenced by:  nmblolbii  28626
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