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Theorem nmcfnlbi 31572
Description: A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcfnex.1 𝑇 ∈ LinFn
nmcfnex.2 𝑇 ∈ ContFn
Assertion
Ref Expression
nmcfnlbi (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
Proof of Theorem nmcfnlbi
StepHypRef Expression
1 fveq2 6890 . . . . . 6 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = (π‘‡β€˜0β„Ž))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 31562 . . . . . 6 (π‘‡β€˜0β„Ž) = 0
41, 3eqtrdi 2786 . . . . 5 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = 0)
54abs00bd 15242 . . . 4 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) = 0)
6 0le0 12317 . . . . 5 0 ≀ 0
7 fveq2 6890 . . . . . . . 8 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = (normβ„Žβ€˜0β„Ž))
8 norm0 30648 . . . . . . . 8 (normβ„Žβ€˜0β„Ž) = 0
97, 8eqtrdi 2786 . . . . . . 7 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = 0)
109oveq2d 7427 . . . . . 6 (𝐴 = 0β„Ž β†’ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)) = ((normfnβ€˜π‘‡) Β· 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 31571 . . . . . . . 8 (normfnβ€˜π‘‡) ∈ ℝ
1312recni 11232 . . . . . . 7 (normfnβ€˜π‘‡) ∈ β„‚
1413mul01i 11408 . . . . . 6 ((normfnβ€˜π‘‡) Β· 0) = 0
1510, 14eqtr2di 2787 . . . . 5 (𝐴 = 0β„Ž β†’ 0 = ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
166, 15breqtrid 5184 . . . 4 (𝐴 = 0β„Ž β†’ 0 ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
175, 16eqbrtrd 5169 . . 3 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
1817adantl 480 . 2 ((𝐴 ∈ β„‹ ∧ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
192lnfnfi 31561 . . . . . . . . . 10 𝑇: β„‹βŸΆβ„‚
2019ffvelcdmi 7084 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
2120abscld 15387 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2221adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2322recnd 11246 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ β„‚)
24 normcl 30645 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2524adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2625recnd 11246 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
27 norm-i 30649 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž))
2827notbid 317 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (Β¬ (normβ„Žβ€˜π΄) = 0 ↔ Β¬ 𝐴 = 0β„Ž))
2928biimpar 476 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ Β¬ (normβ„Žβ€˜π΄) = 0)
3029neqned 2945 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) β‰  0)
3123, 26, 30divrec2d 11998 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) = ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))))
3225, 30rereccld 12045 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
3332recnd 11246 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ β„‚)
34 simpl 481 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 𝐴 ∈ β„‹)
352lnfnmuli 31564 . . . . . . . 8 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ (π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = ((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄)))
3633, 34, 35syl2anc 582 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = ((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄)))
3736fveq2d 6894 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) = (absβ€˜((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄))))
3820adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (π‘‡β€˜π΄) ∈ β„‚)
3933, 38absmuld 15405 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄))) = ((absβ€˜(1 / (normβ„Žβ€˜π΄))) Β· (absβ€˜(π‘‡β€˜π΄))))
40 df-ne 2939 . . . . . . . . . . . 12 (𝐴 β‰  0β„Ž ↔ Β¬ 𝐴 = 0β„Ž)
41 normgt0 30647 . . . . . . . . . . . 12 (𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
4240, 41bitr3id 284 . . . . . . . . . . 11 (𝐴 ∈ β„‹ β†’ (Β¬ 𝐴 = 0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
4342biimpa 475 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 0 < (normβ„Žβ€˜π΄))
4425, 43recgt0d 12152 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
45 0re 11220 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 11306 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ) β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
4745, 46mpan 686 . . . . . . . . 9 ((1 / (normβ„Žβ€˜π΄)) ∈ ℝ β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
4832, 44, 47sylc 65 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄)))
4932, 48absidd 15373 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(1 / (normβ„Žβ€˜π΄))) = (1 / (normβ„Žβ€˜π΄)))
5049oveq1d 7426 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((absβ€˜(1 / (normβ„Žβ€˜π΄))) Β· (absβ€˜(π‘‡β€˜π΄))) = ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))))
5137, 39, 503eqtrrd 2775 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))) = (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))))
5231, 51eqtrd 2770 . . . 4 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) = (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))))
53 hvmulcl 30533 . . . . . 6 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
5433, 34, 53syl2anc 582 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
55 normcl 30645 . . . . . . 7 (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹ β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ∈ ℝ)
5654, 55syl 17 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ∈ ℝ)
57 norm1 30769 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
5840, 57sylan2br 593 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
59 eqle 11320 . . . . . 6 (((normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ∈ ℝ ∧ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
6056, 58, 59syl2anc 582 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
61 nmfnlb 31444 . . . . . 6 ((𝑇: β„‹βŸΆβ„‚ ∧ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹ ∧ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
6219, 61mp3an1 1446 . . . . 5 ((((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹ ∧ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
6354, 60, 62syl2anc 582 . . . 4 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
6452, 63eqbrtrd 5169 . . 3 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) ≀ (normfnβ€˜π‘‡))
6512a1i 11 . . . 4 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normfnβ€˜π‘‡) ∈ ℝ)
66 ledivmul2 12097 . . . 4 (((absβ€˜(π‘‡β€˜π΄)) ∈ ℝ ∧ (normfnβ€˜π‘‡) ∈ ℝ ∧ ((normβ„Žβ€˜π΄) ∈ ℝ ∧ 0 < (normβ„Žβ€˜π΄))) β†’ (((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) ≀ (normfnβ€˜π‘‡) ↔ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄))))
6722, 65, 25, 43, 66syl112anc 1372 . . 3 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) ≀ (normfnβ€˜π‘‡) ↔ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄))))
6864, 67mpbid 231 . 2 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
6918, 68pm2.61dan 809 1 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   class class class wbr 5147  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   Β· cmul 11117   < clt 11252   ≀ cle 11253   / cdiv 11875  abscabs 15185   β„‹chba 30439   Β·β„Ž csm 30441  normβ„Žcno 30443  0β„Žc0v 30444  normfncnmf 30471  ContFnccnfn 30473  LinFnclf 30474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-hilex 30519  ax-hv0cl 30523  ax-hvaddid 30524  ax-hfvmul 30525  ax-hvmulid 30526  ax-hvmulass 30527  ax-hvmul0 30530  ax-hfi 30599  ax-his1 30602  ax-his3 30604  ax-his4 30605
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-hnorm 30488  df-hvsub 30491  df-nmfn 31365  df-cnfn 31367  df-lnfn 31368
This theorem is referenced by:  nmcfnlb  31574
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