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Theorem nmbdfnlbi 31887
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmbdfnlb.1 (𝑇 ∈ LinFn ∧ (normfnβ€˜π‘‡) ∈ ℝ)
Assertion
Ref Expression
nmbdfnlbi (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
Proof of Theorem nmbdfnlbi
StepHypRef Expression
1 fveq2 6902 . . . . . 6 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = (π‘‡β€˜0β„Ž))
2 nmbdfnlb.1 . . . . . . . 8 (𝑇 ∈ LinFn ∧ (normfnβ€˜π‘‡) ∈ ℝ)
32simpli 482 . . . . . . 7 𝑇 ∈ LinFn
43lnfn0i 31880 . . . . . 6 (π‘‡β€˜0β„Ž) = 0
51, 4eqtrdi 2784 . . . . 5 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = 0)
65abs00bd 15280 . . . 4 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) = 0)
7 0le0 12353 . . . . 5 0 ≀ 0
8 fveq2 6902 . . . . . . . 8 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = (normβ„Žβ€˜0β„Ž))
9 norm0 30966 . . . . . . . 8 (normβ„Žβ€˜0β„Ž) = 0
108, 9eqtrdi 2784 . . . . . . 7 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = 0)
1110oveq2d 7442 . . . . . 6 (𝐴 = 0β„Ž β†’ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)) = ((normfnβ€˜π‘‡) Β· 0))
122simpri 484 . . . . . . . 8 (normfnβ€˜π‘‡) ∈ ℝ
1312recni 11268 . . . . . . 7 (normfnβ€˜π‘‡) ∈ β„‚
1413mul01i 11444 . . . . . 6 ((normfnβ€˜π‘‡) Β· 0) = 0
1511, 14eqtr2di 2785 . . . . 5 (𝐴 = 0β„Ž β†’ 0 = ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
167, 15breqtrid 5189 . . . 4 (𝐴 = 0β„Ž β†’ 0 ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
176, 16eqbrtrd 5174 . . 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΄)))
193lnfnfi 31879 . . . . . . . . . 10 𝑇: β„‹βŸΆβ„‚
2019ffvelcdmi 7098 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
2120abscld 15425 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2221adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2322recnd 11282 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ β„‚)
24 normcl 30963 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2524adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2625recnd 11282 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
27 normne0 30968 . . . . . . 7 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0β„Ž))
2827biimpar 476 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) β‰  0)
2923, 26, 28divrec2d 12034 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) = ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))))
3025, 28rereccld 12081 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
3130recnd 11282 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ β„‚)
32 simpl 481 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 𝐴 ∈ β„‹)
333lnfnmuli 31882 . . . . . . . 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΄)))
3431, 32, 33syl2anc 582 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = ((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄)))
3534fveq2d 6906 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) = (absβ€˜((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄))))
3620adantr 479 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (π‘‡β€˜π΄) ∈ β„‚)
3731, 36absmuld 15443 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜((1 / (normβ„Žβ€˜π΄)) Β· (π‘‡β€˜π΄))) = ((absβ€˜(1 / (normβ„Žβ€˜π΄))) Β· (absβ€˜(π‘‡β€˜π΄))))
38 normgt0 30965 . . . . . . . . . . 11 (𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
3938biimpa 475 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (normβ„Žβ€˜π΄))
4025, 39recgt0d 12188 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
41 0re 11256 . . . . . . . . . 10 0 ∈ ℝ
42 ltle 11342 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ) β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
4341, 42mpan 688 . . . . . . . . 9 ((1 / (normβ„Žβ€˜π΄)) ∈ ℝ β†’ (0 < (1 / (normβ„Žβ€˜π΄)) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄))))
4430, 40, 43sylc 65 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 ≀ (1 / (normβ„Žβ€˜π΄)))
4530, 44absidd 15411 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(1 / (normβ„Žβ€˜π΄))) = (1 / (normβ„Žβ€˜π΄)))
4645oveq1d 7441 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((absβ€˜(1 / (normβ„Žβ€˜π΄))) Β· (absβ€˜(π‘‡β€˜π΄))) = ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))))
4735, 37, 463eqtrrd 2773 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β· (absβ€˜(π‘‡β€˜π΄))) = (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))))
4829, 47eqtrd 2768 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) = (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))))
49 hvmulcl 30851 . . . . . 6 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
5031, 32, 49syl2anc 582 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
51 normcl 30963 . . . . . . 7 (((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹ β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ∈ ℝ)
5250, 51syl 17 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ∈ ℝ)
53 norm1 31087 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
54 eqle 11356 . . . . . 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)
5552, 53, 54syl2anc 582 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
56 nmfnlb 31762 . . . . . 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‘‡))
5719, 56mp3an1 1444 . . . . 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‘‡))
5850, 55, 57syl2anc 582 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
5948, 58eqbrtrd 5174 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) ≀ (normfnβ€˜π‘‡))
6012a1i 11 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normfnβ€˜π‘‡) ∈ ℝ)
61 ledivmul2 12133 . . . 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΄))))
6222, 60, 25, 39, 61syl112anc 1371 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (((absβ€˜(π‘‡β€˜π΄)) / (normβ„Žβ€˜π΄)) ≀ (normfnβ€˜π‘‡) ↔ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄))))
6359, 62mpbid 231 . 2 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
6418, 63pm2.61dane 3026 1 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  β„‚cc 11146  β„cr 11147  0cc0 11148  1c1 11149   Β· cmul 11153   < clt 11288   ≀ cle 11289   / cdiv 11911  abscabs 15223   β„‹chba 30757   Β·β„Ž csm 30759  normβ„Žcno 30761  0β„Žc0v 30762  normfncnmf 30789  LinFnclf 30792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225  ax-pre-sup 11226  ax-hilex 30837  ax-hv0cl 30841  ax-hvaddid 30842  ax-hfvmul 30843  ax-hvmulid 30844  ax-hvmul0 30848  ax-hfi 30917  ax-his1 30920  ax-his3 30922  ax-his4 30923
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-sup 9475  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-div 11912  df-nn 12253  df-2 12315  df-3 12316  df-n0 12513  df-z 12599  df-uz 12863  df-rp 13017  df-seq 14009  df-exp 14069  df-cj 15088  df-re 15089  df-im 15090  df-sqrt 15224  df-abs 15225  df-hnorm 30806  df-nmfn 31683  df-lnfn 31686
This theorem is referenced by:  nmbdfnlb  31888
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