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Theorem nmcfnlbi 32071
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 6906 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 32061 . . . . . 6 (𝑇‘0) = 0
41, 3eqtrdi 2793 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
54abs00bd 15330 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
6 0le0 12367 . . . . 5 0 ≤ 0
7 fveq2 6906 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
8 norm0 31147 . . . . . . . 8 (norm‘0) = 0
97, 8eqtrdi 2793 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
109oveq2d 7447 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 32070 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 11275 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 11451 . . . . . 6 ((normfn𝑇) · 0) = 0
1510, 14eqtr2di 2794 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
166, 15breqtrid 5180 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
175, 16eqbrtrd 5165 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 481 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
192lnfnfi 32060 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelcdmi 7103 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 15475 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 480 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 11289 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 31144 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 480 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℝ)
2625recnd 11289 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℂ)
27 norm-i 31148 . . . . . . . . 9 (𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
2827notbid 318 . . . . . . . 8 (𝐴 ∈ ℋ → (¬ (norm𝐴) = 0 ↔ ¬ 𝐴 = 0))
2928biimpar 477 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ¬ (norm𝐴) = 0)
3029neqned 2947 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ≠ 0)
3123, 26, 30divrec2d 12047 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3225, 30rereccld 12094 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℝ)
3332recnd 11289 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℂ)
34 simpl 482 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 𝐴 ∈ ℋ)
352lnfnmuli 32063 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3633, 34, 35syl2anc 584 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3736fveq2d 6910 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3820adantr 480 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇𝐴) ∈ ℂ)
3933, 38absmuld 15493 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
40 df-ne 2941 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
41 normgt0 31146 . . . . . . . . . . . 12 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
4240, 41bitr3id 285 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (¬ 𝐴 = 0 ↔ 0 < (norm𝐴)))
4342biimpa 476 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (norm𝐴))
4425, 43recgt0d 12202 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (1 / (norm𝐴)))
45 0re 11263 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 11349 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4745, 46mpan 690 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4832, 44, 47sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 ≤ (1 / (norm𝐴)))
4932, 48absidd 15461 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
5049oveq1d 7446 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
5137, 39, 503eqtrrd 2782 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
5231, 51eqtrd 2777 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
53 hvmulcl 31032 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5433, 34, 53syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
55 normcl 31144 . . . . . . 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 31268 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
5840, 57sylan2br 595 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
59 eqle 11363 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
6056, 58, 59syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
61 nmfnlb 31943 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6219, 61mp3an1 1450 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6354, 60, 62syl2anc 584 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6452, 63eqbrtrd 5165 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6512a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (normfn𝑇) ∈ ℝ)
66 ledivmul2 12147 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6722, 65, 25, 43, 66syl112anc 1376 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6864, 67mpbid 232 . 2 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6918, 68pm2.61dan 813 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295  cle 11296   / cdiv 11920  abscabs 15273  chba 30938   · csm 30940  normcno 30942  0c0v 30943  normfncnmf 30970  ContFnccnfn 30972  LinFnclf 30973
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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-hilex 31018  ax-hv0cl 31022  ax-hvaddid 31023  ax-hfvmul 31024  ax-hvmulid 31025  ax-hvmulass 31026  ax-hvmul0 31029  ax-hfi 31098  ax-his1 31101  ax-his3 31103  ax-his4 31104
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-pss 3971  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-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  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-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  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-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  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-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-hnorm 30987  df-hvsub 30990  df-nmfn 31864  df-cnfn 31866  df-lnfn 31867
This theorem is referenced by:  nmcfnlb  32073
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