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Theorem nmcfnlbi 31043
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 6846 . . . . . 6 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = (π‘‡β€˜0β„Ž))
2 nmcfnex.1 . . . . . . 7 𝑇 ∈ LinFn
32lnfn0i 31033 . . . . . 6 (π‘‡β€˜0β„Ž) = 0
41, 3eqtrdi 2789 . . . . 5 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = 0)
54abs00bd 15185 . . . 4 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) = 0)
6 0le0 12262 . . . . 5 0 ≀ 0
7 fveq2 6846 . . . . . . . 8 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = (normβ„Žβ€˜0β„Ž))
8 norm0 30119 . . . . . . . 8 (normβ„Žβ€˜0β„Ž) = 0
97, 8eqtrdi 2789 . . . . . . 7 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = 0)
109oveq2d 7377 . . . . . 6 (𝐴 = 0β„Ž β†’ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)) = ((normfnβ€˜π‘‡) Β· 0))
11 nmcfnex.2 . . . . . . . . 9 𝑇 ∈ ContFn
122, 11nmcfnexi 31042 . . . . . . . 8 (normfnβ€˜π‘‡) ∈ ℝ
1312recni 11177 . . . . . . 7 (normfnβ€˜π‘‡) ∈ β„‚
1413mul01i 11353 . . . . . 6 ((normfnβ€˜π‘‡) Β· 0) = 0
1510, 14eqtr2di 2790 . . . . 5 (𝐴 = 0β„Ž β†’ 0 = ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
166, 15breqtrid 5146 . . . 4 (𝐴 = 0β„Ž β†’ 0 ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
175, 16eqbrtrd 5131 . . 3 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
1817adantl 483 . 2 ((𝐴 ∈ β„‹ ∧ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
192lnfnfi 31032 . . . . . . . . . 10 𝑇: β„‹βŸΆβ„‚
2019ffvelcdmi 7038 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
2120abscld 15330 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2221adantr 482 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2322recnd 11191 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ β„‚)
24 normcl 30116 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2524adantr 482 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2625recnd 11191 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
27 norm-i 30120 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) = 0 ↔ 𝐴 = 0β„Ž))
2827notbid 318 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (Β¬ (normβ„Žβ€˜π΄) = 0 ↔ Β¬ 𝐴 = 0β„Ž))
2928biimpar 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ Β¬ (normβ„Žβ€˜π΄) = 0)
3029neqned 2947 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜π΄) β‰  0)
3123, 26, 30divrec2d 11943 . . . . 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 11990 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
3332recnd 11191 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ β„‚)
34 simpl 484 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 𝐴 ∈ β„‹)
352lnfnmuli 31035 . . . . . . . 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 585 . . . . . . 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 6850 . . . . . 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 482 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (π‘‡β€˜π΄) ∈ β„‚)
3933, 38absmuld 15348 . . . . . 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 2941 . . . . . . . . . . . 12 (𝐴 β‰  0β„Ž ↔ Β¬ 𝐴 = 0β„Ž)
41 normgt0 30118 . . . . . . . . . . . 12 (𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
4240, 41bitr3id 285 . . . . . . . . . . 11 (𝐴 ∈ β„‹ β†’ (Β¬ 𝐴 = 0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
4342biimpa 478 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 0 < (normβ„Žβ€˜π΄))
4425, 43recgt0d 12097 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
45 0re 11165 . . . . . . . . . 10 0 ∈ ℝ
46 ltle 11251 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ) β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
4745, 46mpan 689 . . . . . . . . 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 15316 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(1 / (normβ„Žβ€˜π΄))) = (1 / (normβ„Žβ€˜π΄)))
5049oveq1d 7376 . . . . . 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 2778 . . . . 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 2773 . . . 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 30004 . . . . . 6 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
5433, 34, 53syl2anc 585 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
55 normcl 30116 . . . . . . 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 30240 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
5840, 57sylan2br 596 . . . . . 6 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
59 eqle 11265 . . . . . 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 585 . . . . 5 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
61 nmfnlb 30915 . . . . . 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 1449 . . . . 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 585 . . . 4 ((𝐴 ∈ β„‹ ∧ Β¬ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
6452, 63eqbrtrd 5131 . . 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 12042 . . . 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 1375 . . 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 812 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  β„‚cc 11057  β„cr 11058  0cc0 11059  1c1 11060   Β· cmul 11064   < clt 11197   ≀ cle 11198   / cdiv 11820  abscabs 15128   β„‹chba 29910   Β·β„Ž csm 29912  normβ„Žcno 29914  0β„Žc0v 29915  normfncnmf 29942  ContFnccnfn 29944  LinFnclf 29945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-hilex 29990  ax-hv0cl 29994  ax-hvaddid 29995  ax-hfvmul 29996  ax-hvmulid 29997  ax-hvmulass 29998  ax-hvmul0 30001  ax-hfi 30070  ax-his1 30073  ax-his3 30075  ax-his4 30076
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-hnorm 29959  df-hvsub 29962  df-nmfn 30836  df-cnfn 30838  df-lnfn 30839
This theorem is referenced by:  nmcfnlb  31045
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