Theorem nmcfnlbi 32344

Description: A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmcfnex.1	⊢ 𝑇 ∈ LinFn
nmcfnex.2	⊢ 𝑇 ∈ ContFn

Assertion

Ref	Expression
nmcfnlbi	⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Proof of Theorem nmcfnlbi

Step	Hyp	Ref	Expression
1		fveq2 6882	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2		nmcfnex.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32334	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
4	1, 3	eqtrdi 2820	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
5	4	abs00bd 15341	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
6		0le0 12341	. . . . 5 ⊢ 0 ≤ 0
7		fveq2 6882	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
8		norm0 31420	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	eqtrdi 2820	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
10	9	oveq2d 7427	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
11		nmcfnex.2	. . . . . . . . 9 ⊢ 𝑇 ∈ ContFn
12	2, 11	nmcfnexi 32343	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 11222	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 11399	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	10, 14	eqtr2di 2821	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	6, 15	breqtrid 5152	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	5, 16	eqbrtrd 5137	. . 3 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
18	17	adantl 486	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
19	2	lnfnfi 32333	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelcdmi 7079	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 15489	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 11236	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 31417	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 11236	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		norm-i 31421	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ))
28	27	notbid 321	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (¬ (normℎ‘𝐴) = 0 ↔ ¬ 𝐴 = 0ℎ))
29	28	biimpar 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ¬ (normℎ‘𝐴) = 0)
30	29	neqned 2971	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ≠ 0)
31	23, 26, 30	divrec2d 11994	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
32	25, 30	rereccld 12041	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
33	32	recnd 11236	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
34		simpl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 𝐴 ∈ ℋ)
35	2	lnfnmuli 32336	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
36	33, 34, 35	syl2anc 595	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
37	36	fveq2d 6886	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))))
38	20	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘𝐴) ∈ ℂ)
39	33, 38	absmuld 15507	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
40		df-ne 2965	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ)
41		normgt0 31419	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
42	40, 41	bitr3id 288	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ ↔ 0 < (normℎ‘𝐴)))
43	42	biimpa 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (normℎ‘𝐴))
44	25, 43	recgt0d 12148	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
45		0re 11209	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
46		ltle 11297	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
47	45, 46	mpan 702	. . . . . . . . 9 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
48	32, 44, 47	sylc 66	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
49	32, 48	absidd 15473	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
50	49	oveq1d 7426	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
51	37, 39, 50	3eqtrrd 2809	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52	31, 51	eqtrd 2804	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
53		hvmulcl 31305	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54	33, 34, 53	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
55		normcl 31417	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
56	54, 55	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
57		norm1 31541	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
58	40, 57	sylan2br 606	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
59		eqle 11311	. . . . . 6 ⊢ (((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
60	56, 58, 59	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
61		nmfnlb 32216	. . . . . 6 ⊢ ((𝑇: ℋ⟶ℂ ∧ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
62	19, 61	mp3an1 1474	. . . . 5 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
63	54, 60, 62	syl2anc 595	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
64	52, 63	eqbrtrd 5137	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇))
65	12	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normfn‘𝑇) ∈ ℝ)
66		ledivmul2 12093	. . . 4 ⊢ (((abs‘(𝑇‘𝐴)) ∈ ℝ ∧ (normfn‘𝑇) ∈ ℝ ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴))) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇) ↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
67	22, 65, 25, 43, 66	syl112anc 1399	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇) ↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
68	64, 67	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
69	18, 68	pm2.61dan 824	1 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   class class class wbr 5113  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11097  ℝcr 11098  0cc0 11099  1c1 11100   · cmul 11104   < clt 11242   ≤ cle 11243   / cdiv 11870  abscabs 15284   ℋchba 31211   ·_ℎ csm 31213  norm_ℎcno 31215  0_ℎc0v 31216  norm_fncnmf 31243  ContFnccnfn 31245  LinFnclf 31246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177  ax-hilex 31291  ax-hv0cl 31295  ax-hvaddid 31296  ax-hfvmul 31297  ax-hvmulid 31298  ax-hvmulass 31299  ax-hvmul0 31302  ax-hfi 31371  ax-his1 31374  ax-his3 31376  ax-his4 31377

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-rp 13016  df-seq 14037  df-exp 14097  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-hnorm 31260  df-hvsub 31263  df-nmfn 32137  df-cnfn 32139  df-lnfn 32140

This theorem is referenced by:  nmcfnlb  32346