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Theorem nmcoplbi 29217
Description: A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcopex.1 𝑇 ∈ LinOp
nmcopex.2 𝑇 ∈ ContOp
Assertion
Ref Expression
nmcoplbi (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))

Proof of Theorem nmcoplbi
StepHypRef Expression
1 0le0 11322 . . . . 5 0 ≤ 0
21a1i 11 . . . 4 (𝐴 = 0 → 0 ≤ 0)
3 fveq2 6353 . . . . . . 7 (𝐴 = 0 → (𝑇𝐴) = (𝑇‘0))
4 nmcopex.1 . . . . . . . 8 𝑇 ∈ LinOp
54lnop0i 29159 . . . . . . 7 (𝑇‘0) = 0
63, 5syl6eq 2810 . . . . . 6 (𝐴 = 0 → (𝑇𝐴) = 0)
76fveq2d 6357 . . . . 5 (𝐴 = 0 → (norm‘(𝑇𝐴)) = (norm‘0))
8 norm0 28315 . . . . 5 (norm‘0) = 0
97, 8syl6eq 2810 . . . 4 (𝐴 = 0 → (norm‘(𝑇𝐴)) = 0)
10 fveq2 6353 . . . . . . 7 (𝐴 = 0 → (norm𝐴) = (norm‘0))
1110, 8syl6eq 2810 . . . . . 6 (𝐴 = 0 → (norm𝐴) = 0)
1211oveq2d 6830 . . . . 5 (𝐴 = 0 → ((normop𝑇) · (norm𝐴)) = ((normop𝑇) · 0))
13 nmcopex.2 . . . . . . . 8 𝑇 ∈ ContOp
144, 13nmcopexi 29216 . . . . . . 7 (normop𝑇) ∈ ℝ
1514recni 10264 . . . . . 6 (normop𝑇) ∈ ℂ
1615mul01i 10438 . . . . 5 ((normop𝑇) · 0) = 0
1712, 16syl6eq 2810 . . . 4 (𝐴 = 0 → ((normop𝑇) · (norm𝐴)) = 0)
182, 9, 173brtr4d 4836 . . 3 (𝐴 = 0 → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
1918adantl 473 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 = 0) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
20 normcl 28312 . . . . . . . . 9 (𝐴 ∈ ℋ → (norm𝐴) ∈ ℝ)
2120adantr 472 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℝ)
22 normne0 28317 . . . . . . . . 9 (𝐴 ∈ ℋ → ((norm𝐴) ≠ 0 ↔ 𝐴 ≠ 0))
2322biimpar 503 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ≠ 0)
2421, 23rereccld 11064 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℝ)
25 normgt0 28314 . . . . . . . . . 10 (𝐴 ∈ ℋ → (𝐴 ≠ 0 ↔ 0 < (norm𝐴)))
2625biimpa 502 . . . . . . . . 9 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (norm𝐴))
2721, 26recgt0d 11170 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 < (1 / (norm𝐴)))
28 0re 10252 . . . . . . . . 9 0 ∈ ℝ
29 ltle 10338 . . . . . . . . 9 ((0 ∈ ℝ ∧ (1 / (norm𝐴)) ∈ ℝ) → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
3028, 29mpan 708 . . . . . . . 8 ((1 / (norm𝐴)) ∈ ℝ → (0 < (1 / (norm𝐴)) → 0 ≤ (1 / (norm𝐴))))
3124, 27, 30sylc 65 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 0 ≤ (1 / (norm𝐴)))
3224, 31absidd 14380 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (abs‘(1 / (norm𝐴))) = (1 / (norm𝐴)))
3332oveq1d 6829 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))) = ((1 / (norm𝐴)) · (norm‘(𝑇𝐴))))
3424recnd 10280 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (1 / (norm𝐴)) ∈ ℂ)
35 simpl 474 . . . . . . . 8 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℋ)
364lnopmuli 29161 . . . . . . . 8 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3734, 35, 36syl2anc 696 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇‘((1 / (norm𝐴)) · 𝐴)) = ((1 / (norm𝐴)) · (𝑇𝐴)))
3837fveq2d 6357 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = (norm‘((1 / (norm𝐴)) · (𝑇𝐴))))
394lnopfi 29158 . . . . . . . . 9 𝑇: ℋ⟶ ℋ
4039ffvelrni 6522 . . . . . . . 8 (𝐴 ∈ ℋ → (𝑇𝐴) ∈ ℋ)
4140adantr 472 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (𝑇𝐴) ∈ ℋ)
42 norm-iii 28327 . . . . . . 7 (((1 / (norm𝐴)) ∈ ℂ ∧ (𝑇𝐴) ∈ ℋ) → (norm‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))))
4334, 41, 42syl2anc 696 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · (𝑇𝐴))) = ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))))
4438, 43eqtrd 2794 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) = ((abs‘(1 / (norm𝐴))) · (norm‘(𝑇𝐴))))
45 normcl 28312 . . . . . . . . 9 ((𝑇𝐴) ∈ ℋ → (norm‘(𝑇𝐴)) ∈ ℝ)
4640, 45syl 17 . . . . . . . 8 (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ∈ ℝ)
4746adantr 472 . . . . . . 7 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ∈ ℝ)
4847recnd 10280 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ∈ ℂ)
4921recnd 10280 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm𝐴) ∈ ℂ)
5048, 49, 23divrec2d 11017 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) = ((1 / (norm𝐴)) · (norm‘(𝑇𝐴))))
5133, 44, 503eqtr4rd 2805 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) = (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))))
52 hvmulcl 28200 . . . . . 6 (((1 / (norm𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
5334, 35, 52syl2anc 696 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((1 / (norm𝐴)) · 𝐴) ∈ ℋ)
54 normcl 28312 . . . . . . 7 (((1 / (norm𝐴)) · 𝐴) ∈ ℋ → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
5553, 54syl 17 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ)
56 norm1 28436 . . . . . 6 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) = 1)
57 eqle 10351 . . . . . 6 (((norm‘((1 / (norm𝐴)) · 𝐴)) ∈ ℝ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) = 1) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
5855, 56, 57syl2anc 696 . . . . 5 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1)
59 nmoplb 29096 . . . . . 6 ((𝑇: ℋ⟶ ℋ ∧ ((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
6039, 59mp3an1 1560 . . . . 5 ((((1 / (norm𝐴)) · 𝐴) ∈ ℋ ∧ (norm‘((1 / (norm𝐴)) · 𝐴)) ≤ 1) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
6153, 58, 60syl2anc 696 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇‘((1 / (norm𝐴)) · 𝐴))) ≤ (normop𝑇))
6251, 61eqbrtrd 4826 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → ((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇))
6314a1i 11 . . . 4 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (normop𝑇) ∈ ℝ)
64 ledivmul2 11114 . . . 4 (((norm‘(𝑇𝐴)) ∈ ℝ ∧ (normop𝑇) ∈ ℝ ∧ ((norm𝐴) ∈ ℝ ∧ 0 < (norm𝐴))) → (((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇) ↔ (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴))))
6547, 63, 21, 26, 64syl112anc 1481 . . 3 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (((norm‘(𝑇𝐴)) / (norm𝐴)) ≤ (normop𝑇) ↔ (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴))))
6662, 65mpbid 222 . 2 ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0) → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
6719, 66pm2.61dane 3019 1 (𝐴 ∈ ℋ → (norm‘(𝑇𝐴)) ≤ ((normop𝑇) · (norm𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  wf 6045  cfv 6049  (class class class)co 6814  cc 10146  cr 10147  0cc0 10148  1c1 10149   · cmul 10153   < clt 10286  cle 10287   / cdiv 10896  abscabs 14193  chil 28106   · csm 28108  normcno 28110  0c0v 28111  normopcnop 28132  ContOpccop 28133  LinOpclo 28134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-hilex 28186  ax-hfvadd 28187  ax-hvcom 28188  ax-hvass 28189  ax-hv0cl 28190  ax-hvaddid 28191  ax-hfvmul 28192  ax-hvmulid 28193  ax-hvmulass 28194  ax-hvdistr1 28195  ax-hvdistr2 28196  ax-hvmul0 28197  ax-hfi 28266  ax-his1 28269  ax-his2 28270  ax-his3 28271  ax-his4 28272
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-sup 8515  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-grpo 27677  df-gid 27678  df-ablo 27729  df-vc 27744  df-nv 27777  df-va 27780  df-ba 27781  df-sm 27782  df-0v 27783  df-nmcv 27785  df-hnorm 28155  df-hba 28156  df-hvsub 28158  df-nmop 29028  df-cnop 29029  df-lnop 29030
This theorem is referenced by:  nmcoplb  29219  cnlnadjlem2  29257  cnlnadjlem7  29262
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