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Theorem nmcfnlbi 32081
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 6907 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 32071 . . . . . 6 (𝑇‘0) = 0
41, 3eqtrdi 2791 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
54abs00bd 15327 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
6 0le0 12365 . . . . 5 0 ≤ 0
7 fveq2 6907 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
8 norm0 31157 . . . . . . . 8 (norm‘0) = 0
97, 8eqtrdi 2791 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
109oveq2d 7447 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 32080 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 11273 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 11449 . . . . . 6 ((normfn𝑇) · 0) = 0
1510, 14eqtr2di 2792 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
166, 15breqtrid 5185 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
175, 16eqbrtrd 5170 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 481 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
192lnfnfi 32070 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelcdmi 7103 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 15472 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 480 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 11287 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 31154 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 480 . . . . . . 7 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℝ)
2625recnd 11287 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ∈ ℂ)
27 norm-i 31158 . . . . . . . . 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 2945 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (norm𝐴) ≠ 0)
3123, 26, 30divrec2d 12045 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3225, 30rereccld 12092 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℝ)
3332recnd 11287 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (1 / (norm𝐴)) ∈ ℂ)
34 simpl 482 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 𝐴 ∈ ℋ)
352lnfnmuli 32073 . . . . . . . 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 6911 . . . . . 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 15490 . . . . . 6 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
40 df-ne 2939 . . . . . . . . . . . 12 (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
41 normgt0 31156 . . . . . . . . . . . 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 12200 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → 0 < (1 / (norm𝐴)))
45 0re 11261 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 11347 . . . . . . . . . 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 15458 . . . . . . 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 2780 . . . . 5 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
5231, 51eqtrd 2775 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
53 hvmulcl 31042 . . . . . 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 31154 . . . . . . 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 31278 . . . . . . 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 11361 . . . . . 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 31953 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
6219, 61mp3an1 1447 . . . . 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 5170 . . 3 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6512a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0) → (normfn𝑇) ∈ ℝ)
66 ledivmul2 12145 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6722, 65, 25, 43, 66syl112anc 1373 . . 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 1537  wcel 2106  wne 2938   class class class wbr 5148  wf 6559  cfv 6563  (class class class)co 7431  cc 11151  cr 11152  0cc0 11153  1c1 11154   · cmul 11158   < clt 11293  cle 11294   / cdiv 11918  abscabs 15270  chba 30948   · csm 30950  normcno 30952  0c0v 30953  normfncnmf 30980  ContFnccnfn 30982  LinFnclf 30983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-hilex 31028  ax-hv0cl 31032  ax-hvaddid 31033  ax-hfvmul 31034  ax-hvmulid 31035  ax-hvmulass 31036  ax-hvmul0 31039  ax-hfi 31108  ax-his1 31111  ax-his3 31113  ax-his4 31114
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-hnorm 30997  df-hvsub 31000  df-nmfn 31874  df-cnfn 31876  df-lnfn 31877
This theorem is referenced by:  nmcfnlb  32083
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