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Theorem nmbdfnlbi 29753
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 6663 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmbdfnlb.1 . . . . . . . 8 (𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)
32simpli 484 . . . . . . 7 𝑇 ∈ LinFn
43lnfn0i 29746 . . . . . 6 (𝑇‘0) = 0
51, 4syl6eq 2869 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
65abs00bd 14639 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
7 0le0 11726 . . . . 5 0 ≤ 0
8 fveq2 6663 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
9 norm0 28832 . . . . . . . 8 (norm‘0) = 0
108, 9syl6eq 2869 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
1110oveq2d 7161 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
122simpri 486 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 10643 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 10818 . . . . . 6 ((normfn𝑇) · 0) = 0
1511, 14syl6req 2870 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
167, 15breqtrid 5094 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
176, 16eqbrtrd 5079 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 482 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
193lnfnfi 29745 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelrni 6842 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 14784 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 10657 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 28829 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
2625recnd 10657 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
27 normne0 28834 . . . . . . 7 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2827biimpar 478 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
2923, 26, 28divrec2d 11408 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3025, 28rereccld 11455 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
3130recnd 10657 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
32 simpl 483 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
333lnfnmuli 29748 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3431, 32, 33syl2anc 584 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3534fveq2d 6667 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3620adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇𝐴) ∈ ℂ)
3731, 36absmuld 14802 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
38 normgt0 28831 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
3938biimpa 477 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
4025, 39recgt0d 11562 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
41 0re 10631 . . . . . . . . . 10 0 ∈ ℝ
42 ltle 10717 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4341, 42mpan 686 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4430, 40, 43sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
4530, 44absidd 14770 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
4645oveq1d 7160 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
4735, 37, 463eqtrrd 2858 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
4829, 47eqtrd 2853 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
49 hvmulcl 28717 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5031, 32, 49syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
51 normcl 28829 . . . . . . 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 28953 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
54 eqle 10730 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
5552, 53, 54syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
56 nmfnlb 29628 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5719, 56mp3an1 1439 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5850, 55, 57syl2anc 584 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5948, 58eqbrtrd 5079 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6012a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn𝑇) ∈ ℝ)
61 ledivmul2 11507 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6222, 60, 25, 39, 61syl112anc 1366 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6359, 62mpbid 233 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6418, 63pm2.61dane 3101 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   · cmul 10530   < clt 10663  cle 10664   / cdiv 11285  abscabs 14581  chba 28623   · csm 28625  normcno 28627  0c0v 28628  normfncnmf 28655  LinFnclf 28658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hilex 28703  ax-hv0cl 28707  ax-hvaddid 28708  ax-hfvmul 28709  ax-hvmulid 28710  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-hnorm 28672  df-nmfn 29549  df-lnfn 29552
This theorem is referenced by:  nmbdfnlb  29754
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