Theorem nmcoplbi 30291

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 → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Proof of Theorem nmcoplbi

Step	Hyp	Ref	Expression
1		0le0 12004	. . . . 5 ⊢ 0 ≤ 0
2	1	a1i 11	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ 0)
3		fveq2 6756	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
4		nmcopex.1	. . . . . . . 8 ⊢ 𝑇 ∈ LinOp
5	4	lnop0i 30233	. . . . . . 7 ⊢ (𝑇‘0ℎ) = 0ℎ
6	3, 5	eqtrdi 2795	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0ℎ)
7	6	fveq2d 6760	. . . . 5 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = (normℎ‘0ℎ))
8		norm0 29391	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	eqtrdi 2795	. . . 4 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = 0)
10		fveq2 6756	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
11	10, 8	eqtrdi 2795	. . . . . 6 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
12	11	oveq2d 7271	. . . . 5 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘𝑇) · 0))
13		nmcopex.2	. . . . . . . 8 ⊢ 𝑇 ∈ ContOp
14	4, 13	nmcopexi 30290	. . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ
15	14	recni 10920	. . . . . 6 ⊢ (normop‘𝑇) ∈ ℂ
16	15	mul01i 11095	. . . . 5 ⊢ ((normop‘𝑇) · 0) = 0
17	12, 16	eqtrdi 2795	. . . 4 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = 0)
18	2, 9, 17	3brtr4d 5102	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
19	18	adantl 481	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
20		normcl 29388	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
22		normne0 29393	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
23	22	biimpar 477	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
24	21, 23	rereccld 11732	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
25		normgt0 29390	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
26	25	biimpa 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
27	21, 26	recgt0d 11839	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
28		0re 10908	. . . . . . . . 9 ⊢ 0 ∈ ℝ
29		ltle 10994	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
30	28, 29	mpan 686	. . . . . . . 8 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
31	24, 27, 30	sylc 65	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
32	24, 31	absidd 15062	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
33	32	oveq1d 7270	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
34	24	recnd 10934	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
35		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
36	4	lnopmuli 30235	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
37	34, 35, 36	syl2anc 583	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
38	37	fveq2d 6760	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))))
39	4	lnopfi 30232	. . . . . . . . 9 ⊢ 𝑇: ℋ⟶ ℋ
40	39	ffvelrni 6942	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
41	40	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℋ)
42		norm-iii 29403	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ (𝑇‘𝐴) ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
43	34, 41, 42	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
44	38, 43	eqtrd 2778	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
45		normcl 29388	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
46	40, 45	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
48	47	recnd 10934	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℂ)
49	21	recnd 10934	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
50	48, 49, 23	divrec2d 11685	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
51	33, 44, 50	3eqtr4rd 2789	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52		hvmulcl 29276	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
53	34, 35, 52	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54		normcl 29388	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
55	53, 54	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
56		norm1 29512	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
57		eqle 11007	. . . . . 6 ⊢ (((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
58	55, 56, 57	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
59		nmoplb 30170	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
60	39, 59	mp3an1 1446	. . . . 5 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
61	53, 58, 60	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
62	51, 61	eqbrtrd 5092	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇))
63	14	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normop‘𝑇) ∈ ℝ)
64		ledivmul2 11784	. . . 4 ⊢ (((normℎ‘(𝑇‘𝐴)) ∈ ℝ ∧ (normop‘𝑇) ∈ ℝ ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴))) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
65	47, 63, 21, 26, 64	syl112anc 1372	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
66	62, 65	mpbid 231	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
67	19, 66	pm2.61dane 3031	1 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   · cmul 10807   < clt 10940   ≤ cle 10941   / cdiv 11562  abscabs 14873   ℋchba 29182   ·_ℎ csm 29184  norm_ℎcno 29186  0_ℎc0v 29187  norm_opcnop 29208  ContOpccop 29209  LinOpclo 29210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-hilex 29262  ax-hfvadd 29263  ax-hvcom 29264  ax-hvass 29265  ax-hv0cl 29266  ax-hvaddid 29267  ax-hfvmul 29268  ax-hvmulid 29269  ax-hvmulass 29270  ax-hvdistr1 29271  ax-hvdistr2 29272  ax-hvmul0 29273  ax-hfi 29342  ax-his1 29345  ax-his2 29346  ax-his3 29347  ax-his4 29348

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-sup 9131  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-grpo 28756  df-gid 28757  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-nmcv 28863  df-hnorm 29231  df-hba 29232  df-hvsub 29234  df-nmop 30102  df-cnop 30103  df-lnop 30104

This theorem is referenced by:  nmcoplb  30293  cnlnadjlem2  30331  cnlnadjlem7  30336