Theorem nmcfnlbi 32404

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 6881	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2		nmcfnex.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32394	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
4	1, 3	eqtrdi 2814	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
5	4	abs00bd 15338	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
6		0le0 12337	. . . . 5 ⊢ 0 ≤ 0
7		fveq2 6881	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
8		norm0 31480	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
10	9	oveq2d 7426	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
11		nmcfnex.2	. . . . . . . . 9 ⊢ 𝑇 ∈ ContFn
12	2, 11	nmcfnexi 32403	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 11218	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 11395	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	10, 14	eqtr2di 2815	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	6, 15	breqtrid 5148	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	5, 16	eqbrtrd 5133	. . 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 32393	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelcdmi 7078	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 15486	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 11232	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 31477	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 11232	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		norm-i 31481	. . . . . . . . 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 2965	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ≠ 0)
31	23, 26, 30	divrec2d 11990	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
32	25, 30	rereccld 12037	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
33	32	recnd 11232	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
34		simpl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 𝐴 ∈ ℋ)
35	2	lnfnmuli 32396	. . . . . . . 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 6885	. . . . . 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 15504	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
40		df-ne 2959	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ)
41		normgt0 31479	. . . . . . . . . . . 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 12144	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
45		0re 11205	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
46		ltle 11293	. . . . . . . . . 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 15470	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
50	49	oveq1d 7425	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
51	37, 39, 50	3eqtrrd 2803	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52	31, 51	eqtrd 2798	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
53		hvmulcl 31365	. . . . . 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 31477	. . . . . . 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 31601	. . . . . . 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 11307	. . . . . 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 32276	. . . . . 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 1477	. . . . 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 5133	. . 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 12089	. . . 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 1401	. . 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 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℂcc 11093  ℝcr 11094  0cc0 11095  1c1 11096   · cmul 11100   < clt 11238   ≤ cle 11239   / cdiv 11866  abscabs 15281   ℋchba 31271   ·_ℎ csm 31273  norm_ℎcno 31275  0_ℎc0v 31276  norm_fncnmf 31303  ContFnccnfn 31305  LinFnclf 31306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-hilex 31351  ax-hv0cl 31355  ax-hvaddid 31356  ax-hfvmul 31357  ax-hvmulid 31358  ax-hvmulass 31359  ax-hvmul0 31362  ax-hfi 31431  ax-his1 31434  ax-his3 31436  ax-his4 31437

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9398  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-seq 14034  df-exp 14094  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-hnorm 31320  df-hvsub 31323  df-nmfn 32197  df-cnfn 32199  df-lnfn 32200

This theorem is referenced by:  nmcfnlb  32406