HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmbdfnlbi Structured version   Visualization version   GIF version

Theorem nmbdfnlbi 30991
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 ∧ (normfn𝑇) ∈ ℝ)
Assertion
Ref Expression
nmbdfnlbi (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))

Proof of Theorem nmbdfnlbi
StepHypRef Expression
1 fveq2 6842 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
2 nmbdfnlb.1 . . . . . . . 8 (𝑇 ∈ LinFn ∧ (normfn𝑇) ∈ ℝ)
32simpli 484 . . . . . . 7 𝑇 ∈ LinFn
43lnfn0i 30984 . . . . . 6 (𝑇‘0) = 0
51, 4eqtrdi 2792 . . . . 5 (𝐴 = 0 → (𝑇𝐴) = 0)
65abs00bd 15176 . . . 4 (𝐴 = 0 → (abs‘(𝑇𝐴)) = 0)
7 0le0 12254 . . . . 5 0 ≤ 0
8 fveq2 6842 . . . . . . . 8 (𝐴 = 0 → (norm𝐴) = (norm‘0))
9 norm0 30070 . . . . . . . 8 (norm‘0) = 0
108, 9eqtrdi 2792 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = 0)
1110oveq2d 7373 . . . . . 6 (𝐴 = 0 → ((normfn𝑇) · (norm𝐴)) = ((normfn𝑇) · 0))
122simpri 486 . . . . . . . 8 (normfn𝑇) ∈ ℝ
1312recni 11169 . . . . . . 7 (normfn𝑇) ∈ ℂ
1413mul01i 11345 . . . . . 6 ((normfn𝑇) · 0) = 0
1511, 14eqtr2di 2793 . . . . 5 (𝐴 = 0 → 0 = ((normfn𝑇) · (norm𝐴)))
167, 15breqtrid 5142 . . . 4 (𝐴 = 0 → 0 ≤ ((normfn𝑇) · (norm𝐴)))
176, 16eqbrtrd 5127 . . 3 (𝐴 = 0 → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
1817adantl 482 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
193lnfnfi 30983 . . . . . . . . . 10 𝑇: ℋ⟶ℂ
2019ffvelcdmi 7034 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℂ)
2120abscld 15321 . . . . . . . 8 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ∈ ℝ)
2221adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℝ)
2322recnd 11183 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ∈ ℂ)
24 normcl 30067 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2524adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
2625recnd 11183 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
27 normne0 30072 . . . . . . 7 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2827biimpar 478 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
2923, 26, 28divrec2d 11935 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
3025, 28rereccld 11982 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
3130recnd 11183 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
32 simpl 483 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
333lnfnmuli 30986 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3431, 32, 33syl2anc 584 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3534fveq2d 6846 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (abs‘((1 / (norm𝐴)) · (𝑇𝐴))))
3620adantr 481 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇𝐴) ∈ ℂ)
3731, 36absmuld 15339 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))))
38 normgt0 30069 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
3938biimpa 477 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
4025, 39recgt0d 12089 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
41 0re 11157 . . . . . . . . . 10 0 ∈ ℝ
42 ltle 11243 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4341, 42mpan 688 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
4430, 40, 43sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
4530, 44absidd 15307 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
4645oveq1d 7372 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(1 / (norm𝐴))) · (abs‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))))
4735, 37, 463eqtrrd 2781 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (abs‘(𝑇𝐴))) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
4829, 47eqtrd 2776 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) = (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
49 hvmulcl 29955 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5031, 32, 49syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
51 normcl 30067 . . . . . . 7 (((1 / (norm𝐴)) · 𝐴) ∈ ℋ → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
5250, 51syl 17 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
53 norm1 30191 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
54 eqle 11257 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
5552, 53, 54syl2anc 584 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
56 nmfnlb 30866 . . . . . 6 ((𝑇: ℋ⟶ℂ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5719, 56mp3an1 1448 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5850, 55, 57syl2anc 584 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normfn𝑇))
5948, 58eqbrtrd 5127 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇))
6012a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normfn𝑇) ∈ ℝ)
61 ledivmul2 12034 . . . 4 (((abs‘(𝑇𝐴)) ∈ ℝ ∧ (normfn𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6222, 60, 25, 39, 61syl112anc 1374 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((abs‘(𝑇𝐴)) / (norm𝐴)) ≤ (normfn𝑇) ↔ (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴))))
6359, 62mpbid 231 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
6418, 63pm2.61dane 3032 1 (𝐴 ∈ ℋ → (abs‘(𝑇𝐴)) ≤ ((normfn𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  abscabs 15119  chba 29861   · csm 29863  normcno 29865  0c0v 29866  normfncnmf 29893  LinFnclf 29896
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hilex 29941  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his3 30026  ax-his4 30027
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-hnorm 29910  df-nmfn 30787  df-lnfn 30790
This theorem is referenced by:  nmbdfnlb  30992
  Copyright terms: Public domain W3C validator