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Theorem nmcfnlbi 29839
Description: A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcfnex.1 𝑇 ∈ LinFn
nmcfnex.2 𝑇 ∈ ContFn
Assertion
Ref Expression
nmcfnlbi (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Proof of Theorem nmcfnlbi
StepHypRef Expression
1 fveq2 6649 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 29829 . . . . . 6 (𝑇‘0) = 0
41, 3eqtrdi 2852 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
54abs00bd 14647 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
6 0le0 11730 . . . . 5 0 ≤ 0
7 fveq2 6649 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
8 norm0 28915 . . . . . . . 8 (norm‘0) = 0
97, 8eqtrdi 2852 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
109oveq2d 7155 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 29838 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 10648 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 10823 . . . . . 6 ((normfn𝑇) · 0) = 0
1510, 14eqtr2di 2853 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
166, 15breqtrid 5070 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
175, 16eqbrtrd 5055 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 485 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
192lnfnfi 29828 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelrni 6831 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 14792 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 10662 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 28912 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℝ)
2625recnd 10662 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℂ)
27 norm-i 28916 . . . . . . . . 9 (𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
2827notbid 321 . . . . . . . 8 (𝐴 ∈ ℋ → (¬ (norm𝐴) = 0 ↔ ¬ 𝐴 = 0))
2928biimpar 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ¬ (norm𝐴) = 0)
3029neqned 2997 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ≠ 0)
3123, 26, 30divrec2d 11413 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3225, 30rereccld 11460 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℝ)
3332recnd 10662 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℂ)
34 simpl 486 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 𝐴 ∈ ℋ)
352lnfnmuli 29831 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3633, 34, 35syl2anc 587 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3736fveq2d 6653 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3820adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇𝐴) ∈ ℂ)
3933, 38absmuld 14810 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
40 df-ne 2991 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
41 normgt0 28914 . . . . . . . . . . . 12 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
4240, 41bitr3id 288 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (¬ 𝐴 = 0 ↔ 0 < (norm𝐴)))
4342biimpa 480 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (norm𝐴))
4425, 43recgt0d 11567 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (1 / (norm𝐴)))
45 0re 10636 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 10722 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4745, 46mpan 689 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4832, 44, 47sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 ≤ (1 / (norm𝐴)))
4932, 48absidd 14778 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
5049oveq1d 7154 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
5137, 39, 503eqtrrd 2841 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
5231, 51eqtrd 2836 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
53 hvmulcl 28800 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5433, 34, 53syl2anc 587 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
55 normcl 28912 . . . . . . 7 (((1 / (norm𝐴)) · 𝐴) ∈ ℋ → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
5654, 55syl 17 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
57 norm1 29036 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
5840, 57sylan2br 597 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
59 eqle 10735 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
6056, 58, 59syl2anc 587 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
61 nmfnlb 29711 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6219, 61mp3an1 1445 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6354, 60, 62syl2anc 587 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6452, 63eqbrtrd 5055 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6512a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (normfn𝑇) ∈ ℝ)
66 ledivmul2 11512 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6722, 65, 25, 43, 66syl112anc 1371 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6864, 67mpbid 235 . 2 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6918, 68pm2.61dan 812 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990   class class class wbr 5033  wf 6324  cfv 6328  (class class class)co 7139  cc 10528  cr 10529  0cc0 10530  1c1 10531   · cmul 10535   < clt 10668  cle 10669   / cdiv 11290  abscabs 14589  chba 28706   · csm 28708  normcno 28710  0c0v 28711  normfncnmf 28738  ContFnccnfn 28740  LinFnclf 28741
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608  ax-hilex 28786  ax-hv0cl 28790  ax-hvaddid 28791  ax-hfvmul 28792  ax-hvmulid 28793  ax-hvmulass 28794  ax-hvmul0 28797  ax-hfi 28866  ax-his1 28869  ax-his3 28871  ax-his4 28872
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  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-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-seq 13369  df-exp 13430  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-hnorm 28755  df-hvsub 28758  df-nmfn 29632  df-cnfn 29634  df-lnfn 29635
This theorem is referenced by:  nmcfnlb  29841
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