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Theorem nmbdfnlbi 31297
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 6891 . . . . . 6 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = (π‘‡β€˜0β„Ž))
2 nmbdfnlb.1 . . . . . . . 8 (𝑇 ∈ LinFn ∧ (normfnβ€˜π‘‡) ∈ ℝ)
32simpli 484 . . . . . . 7 𝑇 ∈ LinFn
43lnfn0i 31290 . . . . . 6 (π‘‡β€˜0β„Ž) = 0
51, 4eqtrdi 2788 . . . . 5 (𝐴 = 0β„Ž β†’ (π‘‡β€˜π΄) = 0)
65abs00bd 15237 . . . 4 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) = 0)
7 0le0 12312 . . . . 5 0 ≀ 0
8 fveq2 6891 . . . . . . . 8 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = (normβ„Žβ€˜0β„Ž))
9 norm0 30376 . . . . . . . 8 (normβ„Žβ€˜0β„Ž) = 0
108, 9eqtrdi 2788 . . . . . . 7 (𝐴 = 0β„Ž β†’ (normβ„Žβ€˜π΄) = 0)
1110oveq2d 7424 . . . . . 6 (𝐴 = 0β„Ž β†’ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)) = ((normfnβ€˜π‘‡) Β· 0))
122simpri 486 . . . . . . . 8 (normfnβ€˜π‘‡) ∈ ℝ
1312recni 11227 . . . . . . 7 (normfnβ€˜π‘‡) ∈ β„‚
1413mul01i 11403 . . . . . 6 ((normfnβ€˜π‘‡) Β· 0) = 0
1511, 14eqtr2di 2789 . . . . 5 (𝐴 = 0β„Ž β†’ 0 = ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
167, 15breqtrid 5185 . . . 4 (𝐴 = 0β„Ž β†’ 0 ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
176, 16eqbrtrd 5170 . . 3 (𝐴 = 0β„Ž β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
1817adantl 482 . 2 ((𝐴 ∈ β„‹ ∧ 𝐴 = 0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ≀ ((normfnβ€˜π‘‡) Β· (normβ„Žβ€˜π΄)))
193lnfnfi 31289 . . . . . . . . . 10 𝑇: β„‹βŸΆβ„‚
2019ffvelcdmi 7085 . . . . . . . . 9 (𝐴 ∈ β„‹ β†’ (π‘‡β€˜π΄) ∈ β„‚)
2120abscld 15382 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2221adantr 481 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ ℝ)
2322recnd 11241 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜π΄)) ∈ β„‚)
24 normcl 30373 . . . . . . . 8 (𝐴 ∈ β„‹ β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2524adantr 481 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ ℝ)
2625recnd 11241 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) ∈ β„‚)
27 normne0 30378 . . . . . . 7 (𝐴 ∈ β„‹ β†’ ((normβ„Žβ€˜π΄) β‰  0 ↔ 𝐴 β‰  0β„Ž))
2827biimpar 478 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜π΄) β‰  0)
2923, 26, 28divrec2d 11993 . . . . 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 12040 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ ℝ)
3130recnd 11241 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (1 / (normβ„Žβ€˜π΄)) ∈ β„‚)
32 simpl 483 . . . . . . . 8 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 𝐴 ∈ β„‹)
333lnfnmuli 31292 . . . . . . . 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 584 . . . . . . 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 6895 . . . . . 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 481 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (π‘‡β€˜π΄) ∈ β„‚)
3731, 36absmuld 15400 . . . . . 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 30375 . . . . . . . . . . 11 (𝐴 ∈ β„‹ β†’ (𝐴 β‰  0β„Ž ↔ 0 < (normβ„Žβ€˜π΄)))
3938biimpa 477 . . . . . . . . . 10 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (normβ„Žβ€˜π΄))
4025, 39recgt0d 12147 . . . . . . . . 9 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ 0 < (1 / (normβ„Žβ€˜π΄)))
41 0re 11215 . . . . . . . . . 10 0 ∈ ℝ
42 ltle 11301 . . . . . . . . . 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 15368 . . . . . . 7 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(1 / (normβ„Žβ€˜π΄))) = (1 / (normβ„Žβ€˜π΄)))
4645oveq1d 7423 . . . . . 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 2777 . . . . 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 2772 . . . 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 30261 . . . . . 6 (((1 / (normβ„Žβ€˜π΄)) ∈ β„‚ ∧ 𝐴 ∈ β„‹) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
5031, 32, 49syl2anc 584 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ ((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴) ∈ β„‹)
51 normcl 30373 . . . . . . 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 30497 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) = 1)
54 eqle 11315 . . . . . 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 584 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (normβ„Žβ€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴)) ≀ 1)
56 nmfnlb 31172 . . . . . 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 1448 . . . . 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 584 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐴 β‰  0β„Ž) β†’ (absβ€˜(π‘‡β€˜((1 / (normβ„Žβ€˜π΄)) Β·β„Ž 𝐴))) ≀ (normfnβ€˜π‘‡))
5948, 58eqbrtrd 5170 . . 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 12092 . . . 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 1374 . . 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 3029 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  β„‚cc 11107  β„cr 11108  0cc0 11109  1c1 11110   Β· cmul 11114   < clt 11247   ≀ cle 11248   / cdiv 11870  abscabs 15180   β„‹chba 30167   Β·β„Ž csm 30169  normβ„Žcno 30171  0β„Žc0v 30172  normfncnmf 30199  LinFnclf 30202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187  ax-hilex 30247  ax-hv0cl 30251  ax-hvaddid 30252  ax-hfvmul 30253  ax-hvmulid 30254  ax-hvmul0 30258  ax-hfi 30327  ax-his1 30330  ax-his3 30332  ax-his4 30333
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-3 12275  df-n0 12472  df-z 12558  df-uz 12822  df-rp 12974  df-seq 13966  df-exp 14027  df-cj 15045  df-re 15046  df-im 15047  df-sqrt 15181  df-abs 15182  df-hnorm 30216  df-nmfn 31093  df-lnfn 31096
This theorem is referenced by:  nmbdfnlb  31298
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