Theorem nmbdfnlbi 31993

Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nmbdfnlb.1	⊢ (𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ)

Assertion

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

Proof of Theorem nmbdfnlbi

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2		nmbdfnlb.1	. . . . . . . 8 ⊢ (𝑇 ∈ LinFn ∧ (normfn‘𝑇) ∈ ℝ)
3	2	simpli 483	. . . . . . 7 ⊢ 𝑇 ∈ LinFn
4	3	lnfn0i 31986	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
5	1, 4	eqtrdi 2780	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
6	5	abs00bd 15198	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
7		0le0 12229	. . . . 5 ⊢ 0 ≤ 0
8		fveq2 6822	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
9		norm0 31072	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
10	8, 9	eqtrdi 2780	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
11	10	oveq2d 7365	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
12	2	simpri 485	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 11129	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 11306	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	11, 14	eqtr2di 2781	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	7, 15	breqtrid 5129	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	6, 16	eqbrtrd 5114	. . 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	3	lnfnfi 31985	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelcdmi 7017	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 15346	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 11143	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 31069	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 11143	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		normne0 31074	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
28	27	biimpar 477	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
29	23, 26, 28	divrec2d 11904	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
30	25, 28	rereccld 11951	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
31	30	recnd 11143	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
32		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
33	3	lnfnmuli 31988	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
34	31, 32, 33	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
35	34	fveq2d 6826	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))))
36	20	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℂ)
37	31, 36	absmuld 15364	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
38		normgt0 31071	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
39	38	biimpa 476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
40	25, 39	recgt0d 12059	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
41		0re 11117	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
42		ltle 11204	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
43	41, 42	mpan 690	. . . . . . . . 9 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
44	30, 40, 43	sylc 65	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
45	30, 44	absidd 15330	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
46	45	oveq1d 7364	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
47	35, 37, 46	3eqtrrd 2769	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
48	29, 47	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
49		hvmulcl 30957	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
50	31, 32, 49	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
51		normcl 31069	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
52	50, 51	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
53		norm1 31193	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
54		eqle 11218	. . . . . 6 ⊢ (((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
55	52, 53, 54	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
56		nmfnlb 31868	. . . . . 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))
57	19, 56	mp3an1 1450	. . . . 5 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
58	50, 55, 57	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
59	48, 58	eqbrtrd 5114	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇))
60	12	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normfn‘𝑇) ∈ ℝ)
61		ledivmul2 12004	. . . 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))))
62	22, 60, 25, 39, 61	syl112anc 1376	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normfn‘𝑇) ↔ (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴))))
63	59, 62	mpbid 232	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
64	18, 63	pm2.61dane 3012	1 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5092  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010   · cmul 11014   < clt 11149   ≤ cle 11150   / cdiv 11777  abscabs 15141   ℋchba 30863   ·_ℎ csm 30865  norm_ℎcno 30867  0_ℎc0v 30868  norm_fncnmf 30895  LinFnclf 30898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087  ax-hilex 30943  ax-hv0cl 30947  ax-hvaddid 30948  ax-hfvmul 30949  ax-hvmulid 30950  ax-hvmul0 30954  ax-hfi 31023  ax-his1 31026  ax-his3 31028  ax-his4 31029

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-hnorm 30912  df-nmfn 31789  df-lnfn 31792

This theorem is referenced by:  nmbdfnlb  31994