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Theorem nmbdfnlb 28293
 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 6102 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇𝐴) = (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴))
21fveq2d 6107 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (abs‘(𝑇𝐴)) = (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)))
3 fveq2 6103 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn𝑇) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
43oveq1d 6564 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) · (norm𝐴)) = ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))
52, 4breq12d 4596 . . . 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 2676 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇 ∈ LinFn ↔ if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn))
83eleq1d 2672 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) ∈ ℝ ↔ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ))
97, 8anbi12d 743 . . . . 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 2676 . . . . . 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 6103 . . . . . . 7 (( ℋ × {0}) = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn‘( ℋ × {0})) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
1211eleq1d 2672 . . . . . 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 743 . . . . 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 28228 . . . . . 6 ( ℋ × {0}) ∈ LinFn
15 nmfn0 28230 . . . . . . 7 (normfn‘( ℋ × {0})) = 0
16 0re 9919 . . . . . . 7 0 ∈ ℝ
1715, 16eqeltri 2684 . . . . . 6 (normfn‘( ℋ × {0})) ∈ ℝ
1814, 17pm3.2i 470 . . . . 5 (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
199, 13, 18elimhyp 4096 . . . 4 (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)
2019nmbdfnlbi 28292 . . 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 4089 . 2 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
22213impia 1253 1 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  {csn 4125   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   · cmul 9820   ≤ cle 9954  abscabs 13822   ℋchil 27160  normℎcno 27164  normfncnmf 27192  LinFnclf 27195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-hilex 27240  ax-hfvadd 27241  ax-hv0cl 27244  ax-hvaddid 27245  ax-hfvmul 27246  ax-hvmulid 27247  ax-hvmul0 27251  ax-hfi 27320  ax-his1 27323  ax-his3 27325  ax-his4 27326 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-hnorm 27209  df-nmfn 28088  df-lnfn 28091 This theorem is referenced by:  lnfncnbd  28300
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