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Theorem nmcfnlbi 29245
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 6332 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 29235 . . . . . 6 (𝑇‘0) = 0
41, 3syl6eq 2820 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
54abs00bd 14238 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
6 0le0 11311 . . . . 5 0 ≤ 0
7 fveq2 6332 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
8 norm0 28319 . . . . . . . 8 (norm‘0) = 0
97, 8syl6eq 2820 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
109oveq2d 6808 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 29244 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 10253 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 10427 . . . . . 6 ((normfn𝑇) · 0) = 0
1510, 14syl6req 2821 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
166, 15syl5breq 4821 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
175, 16eqbrtrd 4806 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 467 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
192lnfnfi 29234 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelrni 6501 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 14382 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 466 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 10269 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 28316 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 466 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℝ)
2625recnd 10269 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℂ)
27 norm-i 28320 . . . . . . . . 9 (𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
2827notbid 307 . . . . . . . 8 (𝐴 ∈ ℋ → (¬ (norm𝐴) = 0 ↔ ¬ 𝐴 = 0))
2928biimpar 463 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ¬ (norm𝐴) = 0)
3029neqned 2949 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ≠ 0)
3123, 26, 30divrec2d 11006 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3225, 30rereccld 11053 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℝ)
3332recnd 10269 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℂ)
34 simpl 468 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 𝐴 ∈ ℋ)
352lnfnmuli 29237 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3633, 34, 35syl2anc 565 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3736fveq2d 6336 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3820adantr 466 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇𝐴) ∈ ℂ)
3933, 38absmuld 14400 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
40 df-ne 2943 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
41 normgt0 28318 . . . . . . . . . . . 12 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
4240, 41syl5bbr 274 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (¬ 𝐴 = 0 ↔ 0 < (norm𝐴)))
4342biimpa 462 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (norm𝐴))
4425, 43recgt0d 11159 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (1 / (norm𝐴)))
45 0re 10241 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 10327 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4745, 46mpan 662 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4832, 44, 47sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 ≤ (1 / (norm𝐴)))
4932, 48absidd 14368 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
5049oveq1d 6807 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
5137, 39, 503eqtrrd 2809 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
5231, 51eqtrd 2804 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
53 hvmulcl 28204 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5433, 34, 53syl2anc 565 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
55 normcl 28316 . . . . . . 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 28440 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
5840, 57sylan2br 574 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
59 eqle 10340 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
6056, 58, 59syl2anc 565 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
61 nmfnlb 29117 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6219, 61mp3an1 1558 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6354, 60, 62syl2anc 565 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6452, 63eqbrtrd 4806 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6512a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (normfn𝑇) ∈ ℝ)
66 ledivmul2 11103 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6722, 65, 25, 43, 66syl112anc 1479 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6864, 67mpbid 222 . 2 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6918, 68pm2.61dan 796 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942   class class class wbr 4784  wf 6027  cfv 6031  (class class class)co 6792  cc 10135  cr 10136  0cc0 10137  1c1 10138   · cmul 10142   < clt 10275  cle 10276   / cdiv 10885  abscabs 14181  chil 28110   · csm 28112  normcno 28114  0c0v 28115  normfncnmf 28142  ContFnccnfn 28144  LinFnclf 28145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215  ax-hilex 28190  ax-hv0cl 28194  ax-hvaddid 28195  ax-hfvmul 28196  ax-hvmulid 28197  ax-hvmulass 28198  ax-hvmul0 28201  ax-hfi 28270  ax-his1 28273  ax-his3 28275  ax-his4 28276
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-sup 8503  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-rp 12035  df-seq 13008  df-exp 13067  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-hnorm 28159  df-hvsub 28162  df-nmfn 29038  df-cnfn 29040  df-lnfn 29041
This theorem is referenced by:  nmcfnlb  29247
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