Theorem nmcfnlbi 32084

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 6920	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2		nmcfnex.1	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
3	2	lnfn0i 32074	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
4	1, 3	eqtrdi 2796	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
5	4	abs00bd 15340	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
6		0le0 12394	. . . . 5 ⊢ 0 ≤ 0
7		fveq2 6920	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
8		norm0 31160	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	eqtrdi 2796	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
10	9	oveq2d 7464	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
11		nmcfnex.2	. . . . . . . . 9 ⊢ 𝑇 ∈ ContFn
12	2, 11	nmcfnexi 32083	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 11304	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 11480	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	10, 14	eqtr2di 2797	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	6, 15	breqtrid 5203	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	5, 16	eqbrtrd 5188	. . 3 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
18	17	adantl 481	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
19	2	lnfnfi 32073	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelcdmi 7117	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 15485	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 11318	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 31157	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 11318	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		norm-i 31161	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) = 0 ↔ 𝐴 = 0ℎ))
28	27	notbid 318	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (¬ (normℎ‘𝐴) = 0 ↔ ¬ 𝐴 = 0ℎ))
29	28	biimpar 477	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ¬ (normℎ‘𝐴) = 0)
30	29	neqned 2953	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘𝐴) ≠ 0)
31	23, 26, 30	divrec2d 12074	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
32	25, 30	rereccld 12121	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
33	32	recnd 11318	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
34		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 𝐴 ∈ ℋ)
35	2	lnfnmuli 32076	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
36	33, 34, 35	syl2anc 583	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
37	36	fveq2d 6924	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))))
38	20	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (𝑇‘𝐴) ∈ ℂ)
39	33, 38	absmuld 15503	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
40		df-ne 2947	. . . . . . . . . . . 12 ⊢ (𝐴 ≠ 0ℎ ↔ ¬ 𝐴 = 0ℎ)
41		normgt0 31159	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
42	40, 41	bitr3id 285	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (¬ 𝐴 = 0ℎ ↔ 0 < (normℎ‘𝐴)))
43	42	biimpa 476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (normℎ‘𝐴))
44	25, 43	recgt0d 12229	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
45		0re 11292	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
46		ltle 11378	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
47	45, 46	mpan 689	. . . . . . . . 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 15471	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
50	49	oveq1d 7463	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
51	37, 39, 50	3eqtrrd 2785	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52	31, 51	eqtrd 2780	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
53		hvmulcl 31045	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54	33, 34, 53	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
55		normcl 31157	. . . . . . 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 31281	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
58	40, 57	sylan2br 594	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
59		eqle 11392	. . . . . 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 583	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
61		nmfnlb 31956	. . . . . 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 1448	. . . . 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 583	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
64	52, 63	eqbrtrd 5188	. . 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 12174	. . . 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 1374	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇) ↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
68	64, 67	mpbid 232	. 2 ⊢ ((𝐴 ∈ ℋ ∧ ¬ 𝐴 = 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
69	18, 68	pm2.61dan 812	1 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   · cmul 11189   < clt 11324   ≤ cle 11325   / cdiv 11947  abscabs 15283   ℋchba 30951   ·_ℎ csm 30953  norm_ℎcno 30955  0_ℎc0v 30956  norm_fncnmf 30983  ContFnccnfn 30985  LinFnclf 30986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262  ax-hilex 31031  ax-hv0cl 31035  ax-hvaddid 31036  ax-hfvmul 31037  ax-hvmulid 31038  ax-hvmulass 31039  ax-hvmul0 31042  ax-hfi 31111  ax-his1 31114  ax-his3 31116  ax-his4 31117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-sup 9511  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-hnorm 31000  df-hvsub 31003  df-nmfn 31877  df-cnfn 31879  df-lnfn 31880

This theorem is referenced by:  nmcfnlb  32086