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Theorem nmcfnlbi 28101
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 ∈ ConFn
Assertion
Ref Expression
nmcfnlbi (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Proof of Theorem nmcfnlbi
StepHypRef Expression
1 fveq2 6088 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 28091 . . . . . 6 (𝑇‘0) = 0
41, 3syl6eq 2659 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
54abs00bd 13825 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
6 0le0 10957 . . . . 5 0 ≤ 0
7 fveq2 6088 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
8 norm0 27175 . . . . . . . 8 (norm‘0) = 0
97, 8syl6eq 2659 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
109oveq2d 6543 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ConFn
122, 11nmcfnexi 28100 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 9908 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 10077 . . . . . 6 ((normfn𝑇) · 0) = 0
1510, 14syl6req 2660 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
166, 15syl5breq 4614 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
175, 16eqbrtrd 4599 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 480 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
192lnfnfi 28090 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelrni 6251 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 13969 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 9924 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 27172 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℝ)
2625recnd 9924 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℂ)
27 norm-i 27176 . . . . . . . . 9 (𝐴 ∈ ℋ → ((norm𝐴) = 0 ↔ 𝐴 = 0))
2827notbid 306 . . . . . . . 8 (𝐴 ∈ ℋ → (¬ (norm𝐴) = 0 ↔ ¬ 𝐴 = 0))
2928biimpar 500 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ¬ (norm𝐴) = 0)
3029neqned 2788 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ≠ 0)
3123, 26, 30divrec2d 10654 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3225, 30rereccld 10701 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℝ)
3332recnd 9924 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℂ)
34 simpl 471 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 𝐴 ∈ ℋ)
352lnfnmuli 28093 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3633, 34, 35syl2anc 690 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3736fveq2d 6092 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3820adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (𝑇𝐴) ∈ ℂ)
3933, 38absmuld 13987 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
40 df-ne 2781 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
41 normgt0 27174 . . . . . . . . . . . 12 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
4240, 41syl5bbr 272 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (¬ 𝐴 = 0 ↔ 0 < (norm𝐴)))
4342biimpa 499 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (norm𝐴))
4425, 43recgt0d 10807 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (1 / (norm𝐴)))
45 0re 9896 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 9977 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4745, 46mpan 701 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4832, 44, 47sylc 62 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 ≤ (1 / (norm𝐴)))
4932, 48absidd 13955 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
5049oveq1d 6542 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
5137, 39, 503eqtrrd 2648 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
5231, 51eqtrd 2643 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
53 hvmulcl 27060 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5433, 34, 53syl2anc 690 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
55 normcl 27172 . . . . . . 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 27296 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
5840, 57sylan2br 491 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
59 eqle 9990 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
6056, 58, 59syl2anc 690 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
61 nmfnlb 27973 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6219, 61mp3an1 1402 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6354, 60, 62syl2anc 690 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6452, 63eqbrtrd 4599 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6512a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (normfn𝑇) ∈ ℝ)
66 ledivmul2 10751 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6722, 65, 25, 43, 66syl112anc 1321 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6864, 67mpbid 220 . 2 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6918, 68pm2.61dan 827 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  wf 5786  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   · cmul 9797   < clt 9930  cle 9931   / cdiv 10533  abscabs 13768  chil 26966   · csm 26968  normcno 26970  0c0v 26971  normfncnmf 26998  ConFnccnfn 27000  LinFnclf 27001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-hilex 27046  ax-hv0cl 27050  ax-hvaddid 27051  ax-hfvmul 27052  ax-hvmulid 27053  ax-hvmulass 27054  ax-hvmul0 27057  ax-hfi 27126  ax-his1 27129  ax-his3 27131  ax-his4 27132
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-sup 8208  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-seq 12619  df-exp 12678  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-hnorm 27015  df-hvsub 27018  df-nmfn 27894  df-cnfn 27896  df-lnfn 27897
This theorem is referenced by:  nmcfnlb  28103
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