Theorem nmcfnlbi 28911

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 6191	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2		nmcfnex.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 28901	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
4	1, 3	syl6eq 2672	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
5	4	abs00bd 14031	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
6		0le0 11110	. . . . 5 ⊢ 0 ≤ 0
7		fveq2 6191	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
8		norm0 27985	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	syl6eq 2672	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
10	9	oveq2d 6666	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
11		nmcfnex.2	. . . . . . . . 9 ⊢ 𝑇 ∈ ContFn
12	2, 11	nmcfnexi 28910	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 10052	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 10226	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	10, 14	syl6req 2673	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	6, 15	syl5breq 4690	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	5, 16	eqbrtrd 4675	. . 3 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
18	17	adantl 482	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
19	2	lnfnfi 28900	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelrni 6358	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 14175	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 10068	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 27982	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 10068	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		norm-i 27986	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ))
28	27	notbid 308	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (¬ (normℎ‘𝐴) = 0 ↔ ¬ 𝐴 = 0ℎ))
29	28	biimpar 502	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ¬ (normℎ‘𝐴) = 0)
30	29	neqned 2801	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ≠ 0)
31	23, 26, 30	divrec2d 10805	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
32	25, 30	rereccld 10852	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
33	32	recnd 10068	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
34		simpl 473	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 𝐴 ∈ ℋ)
35	2	lnfnmuli 28903	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
36	33, 34, 35	syl2anc 693	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
37	36	fveq2d 6195	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))))
38	20	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘𝐴) ∈ ℂ)
39	33, 38	absmuld 14193	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
40		df-ne 2795	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ)
41		normgt0 27984	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
42	40, 41	syl5bbr 274	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ ↔ 0 < (normℎ‘𝐴)))
43	42	biimpa 501	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (normℎ‘𝐴))
44	25, 43	recgt0d 10958	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
45		0re 10040	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
46		ltle 10126	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
47	45, 46	mpan 706	. . . . . . . . 9 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
48	32, 44, 47	sylc 65	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
49	32, 48	absidd 14161	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
50	49	oveq1d 6665	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
51	37, 39, 50	3eqtrrd 2661	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52	31, 51	eqtrd 2656	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
53		hvmulcl 27870	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54	33, 34, 53	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
55		normcl 27982	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
56	54, 55	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
57		norm1 28106	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
58	40, 57	sylan2br 493	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
59		eqle 10139	. . . . . 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 693	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
61		nmfnlb 28783	. . . . . 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 1411	. . . . 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 693	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
64	52, 63	eqbrtrd 4675	. . 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 10902	. . . 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 1330	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇) ↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
68	64, 67	mpbid 222	. 2 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
69	18, 68	pm2.61dan 832	1 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   · cmul 9941   < clt 10074   ≤ cle 10075   / cdiv 10684  abscabs 13974   ℋchil 27776   ·_ℎ csm 27778  norm_ℎcno 27780  0_ℎc0v 27781  norm_fncnmf 27808  ContFnccnfn 27810  LinFnclf 27811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-hilex 27856  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-hnorm 27825  df-hvsub 27828  df-nmfn 28704  df-cnfn 28706  df-lnfn 28707

This theorem is referenced by:  nmcfnlb  28913