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Theorem nmbdfnlb 29216
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 6349 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (𝑇𝐴) = (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴))
21fveq2d 6354 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (abs‘(𝑇𝐴)) = (abs‘(if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))‘𝐴)))
3 fveq2 6350 . . . . . 6 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → (normfn𝑇) = (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))))
43oveq1d 6826 . . . . 5 (𝑇 = if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) → ((normfn𝑇) · (norm𝐴)) = ((normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) · (norm𝐴)))
52, 4breq12d 4815 . . . 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 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 749 . . . . 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 6350 . . . . . . 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 749 . . . . 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 29151 . . . . . 6 ( ℋ × {0}) ∈ LinFn
15 nmfn0 29153 . . . . . . 7 (normfn‘( ℋ × {0})) = 0
16 0re 10230 . . . . . . 7 0 ∈ ℝ
1715, 16eqeltri 2833 . . . . . 6 (normfn‘( ℋ × {0})) ∈ ℝ
1814, 17pm3.2i 470 . . . . 5 (( ℋ × {0}) ∈ LinFn ∧ (normfn‘( ℋ × {0})) ∈ ℝ)
199, 13, 18elimhyp 4288 . . . 4 (if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0})) ∈ LinFn ∧ (normfn‘if((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ), 𝑇, ( ℋ × {0}))) ∈ ℝ)
2019nmbdfnlbi 29215 . . 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 4281 . 2 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ) → (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
22213impia 1110 1 ((𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ ∧ 𝐴 ∈ ℋ) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  ifcif 4228  {csn 4319   class class class wbr 4802   × cxp 5262  cfv 6047  (class class class)co 6811  cr 10125  0cc0 10126   · cmul 10131  cle 10265  abscabs 14171  chil 28083  normcno 28087  normfncnmf 28115  LinFnclf 28118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202  ax-pre-mulgt0 10203  ax-pre-sup 10204  ax-hilex 28163  ax-hfvadd 28164  ax-hv0cl 28167  ax-hvaddid 28168  ax-hfvmul 28169  ax-hvmulid 28170  ax-hvmul0 28174  ax-hfi 28243  ax-his1 28246  ax-his3 28248  ax-his4 28249
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-sup 8511  df-pnf 10266  df-mnf 10267  df-xr 10268  df-ltxr 10269  df-le 10270  df-sub 10458  df-neg 10459  df-div 10875  df-nn 11211  df-2 11269  df-3 11270  df-n0 11483  df-z 11568  df-uz 11878  df-rp 12024  df-seq 12994  df-exp 13053  df-cj 14036  df-re 14037  df-im 14038  df-sqrt 14172  df-abs 14173  df-hnorm 28132  df-nmfn 29011  df-lnfn 29014
This theorem is referenced by:  lnfncnbd  29223
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