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Theorem nmbdfnlb 30131
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 6716 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇𝐴) = (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴))
21fveq2d 6721 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (abs‘(𝑇𝐴)) = (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)))
3 fveq2 6717 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn𝑇) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
43oveq1d 7228 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) · (norm𝐴)) = ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))
52, 4breq12d 5066 . . . 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 344 . . 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 2825 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇 ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
83eleq1d 2822 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
97, 8anbi12d 634 . . . . 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 2825 . . . . . 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 6717 . . . . . . 7 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn‘( ℋ × {0})) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
1211eleq1d 2822 . . . . . 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 634 . . . . 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 30066 . . . . . 6 ( ℋ × {0}) ∈ LinFn
15 nmfn0 30068 . . . . . . 7 (normfn‘( ℋ × {0})) = 0
16 0re 10835 . . . . . . 7 0 ∈ ℝ
1715, 16eqeltri 2834 . . . . . 6 (normfn‘( ℋ × {0})) ∈ ℝ
1814, 17pm3.2i 474 . . . . 5 (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
199, 13, 18elimhyp 4504 . . . 4 (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)
2019nmbdfnlbi 30130 . . 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 4497 . 2 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
22213impia 1119 1 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  ifcif 4439  {csn 4541   class class class wbr 5053   × cxp 5549  cfv 6380  (class class class)co 7213  cr 10728  0cc0 10729   · cmul 10734  cle 10868  abscabs 14797  chba 29000  normcno 29004  normfncnmf 29032  LinFnclf 29035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-hilex 29080  ax-hfvadd 29081  ax-hv0cl 29084  ax-hvaddid 29085  ax-hfvmul 29086  ax-hvmulid 29087  ax-hvmul0 29091  ax-hfi 29160  ax-his1 29163  ax-his3 29165  ax-his4 29166
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-hnorm 29049  df-nmfn 29926  df-lnfn 29929
This theorem is referenced by:  lnfncnbd  30138
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