Theorem nmbdfnlbi 32120

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 6840	. . . . . 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 32113	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0
5	1, 4	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0)
6	5	abs00bd 15253	. . . 4 ⊢ (𝐴 = 0ℎ → (abs‘(𝑇‘𝐴)) = 0)
7		0le0 12282	. . . . 5 ⊢ 0 ≤ 0
8		fveq2 6840	. . . . . . . 8 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
9		norm0 31199	. . . . . . . 8 ⊢ (normℎ‘0ℎ) = 0
10	8, 9	eqtrdi 2787	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
11	10	oveq2d 7383	. . . . . 6 ⊢ (𝐴 = 0ℎ → ((normfn‘𝑇) · (normℎ‘𝐴)) = ((normfn‘𝑇) · 0))
12	2	simpri 485	. . . . . . . 8 ⊢ (normfn‘𝑇) ∈ ℝ
13	12	recni 11159	. . . . . . 7 ⊢ (normfn‘𝑇) ∈ ℂ
14	13	mul01i 11336	. . . . . 6 ⊢ ((normfn‘𝑇) · 0) = 0
15	11, 14	eqtr2di 2788	. . . . 5 ⊢ (𝐴 = 0ℎ → 0 = ((normfn‘𝑇) · (normℎ‘𝐴)))
16	7, 15	breqtrid 5122	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))
17	6, 16	eqbrtrd 5107	. . 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 32112	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ℂ
20	19	ffvelcdmi 7035	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ)
21	20	abscld 15401	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ∈ ℝ)
22	21	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℝ)
23	22	recnd 11173	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘𝐴)) ∈ ℂ)
24		normcl 31196	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
26	25	recnd 11173	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
27		normne0 31201	. . . . . . 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 11935	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
30	25, 28	rereccld 11982	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
31	30	recnd 11173	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
32		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
33	3	lnfnmuli 32115	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
34	31, 32, 33	syl2anc 585	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) · (𝑇‘𝐴)))
35	34	fveq2d 6844	. . . . . 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 15419	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘((1 / (normℎ‘𝐴)) · (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))))
38		normgt0 31198	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
39	38	biimpa 476	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
40	25, 39	recgt0d 12090	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
41		0re 11146	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
42		ltle 11234	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
43	41, 42	mpan 691	. . . . . . . . 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 15385	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
46	45	oveq1d 7382	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (abs‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))))
47	35, 37, 46	3eqtrrd 2776	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (abs‘(𝑇‘𝐴))) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
48	29, 47	eqtrd 2771	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
49		hvmulcl 31084	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
50	31, 32, 49	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
51		normcl 31196	. . . . . . 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 31320	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
54		eqle 11248	. . . . . 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 585	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
56		nmfnlb 31995	. . . . . 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 1451	. . . . 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 585	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normfn‘𝑇))
59	48, 58	eqbrtrd 5107	. . 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 12035	. . . 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 1377	. . 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 3019	1 ⊢ (𝐴 ∈ ℋ → (abs‘(𝑇‘𝐴)) ≤ ((normfn‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   class class class wbr 5085  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   · cmul 11043   < clt 11179   ≤ cle 11180   / cdiv 11807  abscabs 15196   ℋchba 30990   ·_ℎ csm 30992  norm_ℎcno 30994  0_ℎc0v 30995  norm_fncnmf 31022  LinFnclf 31025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-hilex 31070  ax-hv0cl 31074  ax-hvaddid 31075  ax-hfvmul 31076  ax-hvmulid 31077  ax-hvmul0 31081  ax-hfi 31150  ax-his1 31153  ax-his3 31155  ax-his4 31156

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-sup 9355  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-seq 13964  df-exp 14024  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-hnorm 31039  df-nmfn 31916  df-lnfn 31919

This theorem is referenced by:  nmbdfnlb  32121