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Theorem nmbdfnlb 29037
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmbdfnlb ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Proof of Theorem nmbdfnlb
StepHypRef Expression
1 fveq1 6228 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇𝐴) = (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴))
21fveq2d 6233 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (abs‘(𝑇𝐴)) = (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)))
3 fveq2 6229 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn𝑇) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
43oveq1d 6705 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) · (norm𝐴)) = ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))
52, 4breq12d 4698 . . . 4 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)) ↔ (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴))))
65imbi2d 329 . . 3 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))) ↔ (𝐴 ∈ ℋ → (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))))
7 eleq1 2718 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇 ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
83eleq1d 2715 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
97, 8anbi12d 747 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) ↔ (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)))
10 eleq1 2718 . . . . . 6 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (( ℋ × {0}) ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
11 fveq2 6229 . . . . . . 7 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn‘( ℋ × {0})) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
1211eleq1d 2715 . . . . . 6 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn‘( ℋ × {0})) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
1310, 12anbi12d 747 . . . . 5 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ) ↔ (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)))
14 0lnfn 28972 . . . . . 6 ( ℋ × {0}) ∈ LinFn
15 nmfn0 28974 . . . . . . 7 (normfn‘( ℋ × {0})) = 0
16 0re 10078 . . . . . . 7 0 ∈ ℝ
1715, 16eqeltri 2726 . . . . . 6 (normfn‘( ℋ × {0})) ∈ ℝ
1814, 17pm3.2i 470 . . . . 5 (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
199, 13, 18elimhyp 4179 . . . 4 (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)
2019nmbdfnlbi 29036 . . 3 (𝐴 ∈ ℋ → (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)) ≤ ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))
216, 20dedth 4172 . 2 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
22213impia 1280 1 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  ifcif 4119  {csn 4210   class class class wbr 4685   × cxp 5141  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   · cmul 9979  cle 10113  abscabs 14018  chil 27904  normcno 27908  normfncnmf 27936  LinFnclf 27939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-hilex 27984  ax-hfvadd 27985  ax-hv0cl 27988  ax-hvaddid 27989  ax-hfvmul 27990  ax-hvmulid 27991  ax-hvmul0 27995  ax-hfi 28064  ax-his1 28067  ax-his3 28069  ax-his4 28070
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-hnorm 27953  df-nmfn 28832  df-lnfn 28835
This theorem is referenced by:  lnfncnbd  29044
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