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Theorem nmbdfnlbi 31979
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmbdfnlb.1 (𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)
Assertion
Ref Expression
nmbdfnlbi (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Proof of Theorem nmbdfnlbi
StepHypRef Expression
1 fveq2 6893 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmbdfnlb.1 . . . . . . . 8 (𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)
32simpli 482 . . . . . . 7 𝑇 ∈ LinFn
43lnfn0i 31972 . . . . . 6 (𝑇‘0) = 0
51, 4eqtrdi 2782 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
65abs00bd 15291 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
7 0le0 12359 . . . . 5 0 ≤ 0
8 fveq2 6893 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
9 norm0 31058 . . . . . . . 8 (norm‘0) = 0
108, 9eqtrdi 2782 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
1110oveq2d 7432 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
122simpri 484 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 11269 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 11445 . . . . . 6 ((normfn𝑇) · 0) = 0
1511, 14eqtr2di 2783 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
167, 15breqtrid 5182 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
176, 16eqbrtrd 5167 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 480 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
193lnfnfi 31971 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelcdmi 7089 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 15436 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 11283 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 31055 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
2625recnd 11283 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
27 normne0 31060 . . . . . . 7 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2827biimpar 476 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
2923, 26, 28divrec2d 12039 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3025, 28rereccld 12086 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
3130recnd 11283 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
32 simpl 481 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
333lnfnmuli 31974 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3431, 32, 33syl2anc 582 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3534fveq2d 6897 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3620adantr 479 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇𝐴) ∈ ℂ)
3731, 36absmuld 15454 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
38 normgt0 31057 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
3938biimpa 475 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
4025, 39recgt0d 12194 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
41 0re 11257 . . . . . . . . . 10 0 ∈ ℝ
42 ltle 11343 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4341, 42mpan 688 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4430, 40, 43sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
4530, 44absidd 15422 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
4645oveq1d 7431 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
4735, 37, 463eqtrrd 2771 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
4829, 47eqtrd 2766 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
49 hvmulcl 30943 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5031, 32, 49syl2anc 582 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
51 normcl 31055 . . . . . . 7 (((1 / (norm𝐴)) · 𝐴) ∈ ℋ → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
5250, 51syl 17 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
53 norm1 31179 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
54 eqle 11357 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
5552, 53, 54syl2anc 582 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
56 nmfnlb 31854 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5719, 56mp3an1 1445 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5850, 55, 57syl2anc 582 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5948, 58eqbrtrd 5167 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6012a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn𝑇) ∈ ℝ)
61 ledivmul2 12139 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6222, 60, 25, 39, 61syl112anc 1371 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6359, 62mpbid 231 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6418, 63pm2.61dane 3019 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wne 2930   class class class wbr 5145  wf 6542  cfv 6546  (class class class)co 7416  cc 11147  cr 11148  0cc0 11149  1c1 11150   · cmul 11154   < clt 11289  cle 11290   / cdiv 11912  abscabs 15234  chba 30849   · csm 30851  normcno 30853  0c0v 30854  normfncnmf 30881  LinFnclf 30884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-pre-sup 11227  ax-hilex 30929  ax-hv0cl 30933  ax-hvaddid 30934  ax-hfvmul 30935  ax-hvmulid 30936  ax-hvmul0 30940  ax-hfi 31009  ax-his1 31012  ax-his3 31014  ax-his4 31015
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-sup 9478  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-div 11913  df-nn 12259  df-2 12321  df-3 12322  df-n0 12519  df-z 12605  df-uz 12869  df-rp 13023  df-seq 14016  df-exp 14076  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-hnorm 30898  df-nmfn 31775  df-lnfn 31778
This theorem is referenced by:  nmbdfnlb  31980
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