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Theorem nmbdoplbi 29810
Description: A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmbdoplb.1 𝑇 ∈ BndLinOp
Assertion
Ref Expression
nmbdoplbi (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Proof of Theorem nmbdoplbi
StepHypRef Expression
1 fveq2 6661 . . . 4 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
21fveq2d 6665 . . 3 (𝐴 = 0 → (norm‘(𝑇𝐴)) = (norm‘(𝑇‘0)))
3 fveq2 6661 . . . 4 (𝐴 = 0 → (norm𝐴) = (norm‘0))
43oveq2d 7165 . . 3 (𝐴 = 0 → ((normop𝑇) · (norm𝐴)) = ((normop𝑇) · (norm‘0)))
52, 4breq12d 5065 . 2 (𝐴 = 0 → ((norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)) ↔ (norm‘(𝑇‘0)) ≤ ((normop𝑇) · (norm‘0))))
6 nmbdoplb.1 . . . . . . . . . . . 12 𝑇 ∈ BndLinOp
7 bdopln 29647 . . . . . . . . . . . 12 (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
86, 7ax-mp 5 . . . . . . . . . . 11 𝑇 ∈ LinOp
98lnopfi 29755 . . . . . . . . . 10 𝑇: ℋ⟶ ℋ
109ffvelrni 6841 . . . . . . . . 9 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
11 normcl 28911 . . . . . . . . 9 ((𝑇𝐴) ∈ ℋ → (norm‘(𝑇𝐴)) ∈ ℝ)
1210, 11syl 17 . . . . . . . 8 (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ∈ ℝ)
1312adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ∈ ℝ)
1413recnd 10667 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ∈ ℂ)
15 normcl 28911 . . . . . . . 8 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
1615adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
1716recnd 10667 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
18 normne0 28916 . . . . . . 7 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
1918biimpar 481 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
2014, 17, 19divrec2d 11418 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (norm‘(𝑇𝐴))))
2116, 19rereccld 11465 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
2221recnd 10667 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
23 simpl 486 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
248lnopmuli 29758 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
2522, 23, 24syl2anc 587 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
2625fveq2d 6665 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (norm‘((1 / (norm𝐴)) · (𝑇𝐴))))
2710adantr 484 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇𝐴) ∈ ℋ)
28 norm-iii 28926 . . . . . . 7 (((1 / (norm𝐴)) ∈ ℂ ∧ (𝑇𝐴) ∈ ℋ) → (norm‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))))
2922, 27, 28syl2anc 587 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))))
30 normgt0 28913 . . . . . . . . . . 11 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
3130biimpa 480 . . . . . . . . . 10 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
3216, 31recgt0d 11572 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
33 0re 10641 . . . . . . . . . 10 0 ∈ ℝ
34 ltle 10727 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
3533, 34mpan 689 . . . . . . . . 9 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
3621, 32, 35sylc 65 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
3721, 36absidd 14782 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
3837oveq1d 7164 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (norm‘(𝑇𝐴))))
3926, 29, 383eqtrrd 2864 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · (norm‘(𝑇𝐴))) = (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
4020, 39eqtrd 2859 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) = (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
41 hvmulcl 28799 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
4222, 23, 41syl2anc 587 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
43 normcl 28911 . . . . . . 7 (((1 / (norm𝐴)) · 𝐴) ∈ ℋ → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
4442, 43syl 17 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
45 norm1 29035 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
46 eqle 10740 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
4744, 45, 46syl2anc 587 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
48 nmoplb 29693 . . . . . 6 ((𝑇: ℋ⟶ ℋ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
499, 48mp3an1 1445 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
5042, 47, 49syl2anc 587 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
5140, 50eqbrtrd 5074 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇))
52 nmopre 29656 . . . . . 6 (𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)
536, 52ax-mp 5 . . . . 5 (normop𝑇) ∈ ℝ
5453a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normop𝑇) ∈ ℝ)
55 ledivmul2 11517 . . . 4 (((norm‘(𝑇𝐴)) ∈ ℝ ∧ (normop𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇) ↔ (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴))))
5613, 54, 16, 31, 55syl112anc 1371 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇) ↔ (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴))))
5751, 56mpbid 235 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
58 0le0 11735 . . . 4 0 ≤ 0
598lnop0i 29756 . . . . . 6 (𝑇‘0) = 0
6059fveq2i 6664 . . . . 5 (norm‘(𝑇‘0)) = (norm‘0)
61 norm0 28914 . . . . 5 (norm‘0) = 0
6260, 61eqtri 2847 . . . 4 (norm‘(𝑇‘0)) = 0
6361oveq2i 7160 . . . . 5 ((normop𝑇) · (norm‘0)) = ((normop𝑇) · 0)
6453recni 10653 . . . . . 6 (normop𝑇) ∈ ℂ
6564mul01i 10828 . . . . 5 ((normop𝑇) · 0) = 0
6663, 65eqtri 2847 . . . 4 ((normop𝑇) · (norm‘0)) = 0
6758, 62, 663brtr4i 5082 . . 3 (norm‘(𝑇‘0)) ≤ ((normop𝑇) · (norm‘0))
6867a1i 11 . 2 (𝐴 ∈ ℋ → (norm‘(𝑇‘0)) ≤ ((normop𝑇) · (norm‘0)))
695, 57, 68pm2.61ne 3099 1 (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  wf 6339  cfv 6343  (class class class)co 7149  cc 10533  cr 10534  0cc0 10535  1c1 10536   · cmul 10540   < clt 10673  cle 10674   / cdiv 11295  abscabs 14593  chba 28705   · csm 28707  normcno 28709  0c0v 28710  normopcnop 28731  LinOpclo 28733  BndLinOpcbo 28734
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-hilex 28785  ax-hfvadd 28786  ax-hvcom 28787  ax-hvass 28788  ax-hv0cl 28789  ax-hvaddid 28790  ax-hfvmul 28791  ax-hvmulid 28792  ax-hvmulass 28793  ax-hvdistr1 28794  ax-hvdistr2 28795  ax-hvmul0 28796  ax-hfi 28865  ax-his1 28868  ax-his2 28869  ax-his3 28870  ax-his4 28871
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-sup 8903  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-seq 13374  df-exp 13435  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-grpo 28279  df-gid 28280  df-ablo 28331  df-vc 28345  df-nv 28378  df-va 28381  df-ba 28382  df-sm 28383  df-0v 28384  df-nmcv 28386  df-hnorm 28754  df-hba 28755  df-hvsub 28757  df-nmop 29625  df-lnop 29627  df-bdop 29628
This theorem is referenced by:  nmbdoplb  29811  nmopcoadji  29887
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