Theorem nmcoplbi 29807

Description: A lower bound for the norm of a continuous linear operator. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 7-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nmcopex.1	⊢ 𝑇 ∈ LinOp
nmcopex.2	⊢ 𝑇 ∈ ContOp

Assertion

Ref	Expression
nmcoplbi	⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Proof of Theorem nmcoplbi

Step	Hyp	Ref	Expression
1		0le0 11741	. . . . 5 ⊢ 0 ≤ 0
2	1	a1i 11	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ 0)
3		fveq2 6672	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
4		nmcopex.1	. . . . . . . 8 ⊢ 𝑇 ∈ LinOp
5	4	lnop0i 29749	. . . . . . 7 ⊢ (𝑇‘0ℎ) = 0ℎ
6	3, 5	syl6eq 2874	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0ℎ)
7	6	fveq2d 6676	. . . . 5 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = (normℎ‘0ℎ))
8		norm0 28907	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	syl6eq 2874	. . . 4 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = 0)
10		fveq2 6672	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
11	10, 8	syl6eq 2874	. . . . . 6 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
12	11	oveq2d 7174	. . . . 5 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘𝑇) · 0))
13		nmcopex.2	. . . . . . . 8 ⊢ 𝑇 ∈ ContOp
14	4, 13	nmcopexi 29806	. . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ
15	14	recni 10657	. . . . . 6 ⊢ (normop‘𝑇) ∈ ℂ
16	15	mul01i 10832	. . . . 5 ⊢ ((normop‘𝑇) · 0) = 0
17	12, 16	syl6eq 2874	. . . 4 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = 0)
18	2, 9, 17	3brtr4d 5100	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
19	18	adantl 484	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
20		normcl 28904	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
21	20	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
22		normne0 28909	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
23	22	biimpar 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
24	21, 23	rereccld 11469	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
25		normgt0 28906	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
26	25	biimpa 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
27	21, 26	recgt0d 11576	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
28		0re 10645	. . . . . . . . 9 ⊢ 0 ∈ ℝ
29		ltle 10731	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
30	28, 29	mpan 688	. . . . . . . 8 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
31	24, 27, 30	sylc 65	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
32	24, 31	absidd 14784	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
33	32	oveq1d 7173	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
34	24	recnd 10671	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
35		simpl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
36	4	lnopmuli 29751	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
37	34, 35, 36	syl2anc 586	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
38	37	fveq2d 6676	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))))
39	4	lnopfi 29748	. . . . . . . . 9 ⊢ 𝑇: ℋ⟶ ℋ
40	39	ffvelrni 6852	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
41	40	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℋ)
42		norm-iii 28919	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ (𝑇‘𝐴) ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
43	34, 41, 42	syl2anc 586	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
44	38, 43	eqtrd 2858	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
45		normcl 28904	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
46	40, 45	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
47	46	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
48	47	recnd 10671	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℂ)
49	21	recnd 10671	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
50	48, 49, 23	divrec2d 11422	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
51	33, 44, 50	3eqtr4rd 2869	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52		hvmulcl 28792	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
53	34, 35, 52	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54		normcl 28904	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
55	53, 54	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
56		norm1 29028	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
57		eqle 10744	. . . . . 6 ⊢ (((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
58	55, 56, 57	syl2anc 586	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
59		nmoplb 29686	. . . . . 6 ⊢ ((𝑇: ℋ⟶ ℋ ∧ ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
60	39, 59	mp3an1 1444	. . . . 5 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
61	53, 58, 60	syl2anc 586	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
62	51, 61	eqbrtrd 5090	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇))
63	14	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normop‘𝑇) ∈ ℝ)
64		ledivmul2 11521	. . . 4 ⊢ (((normℎ‘(𝑇‘𝐴)) ∈ ℝ ∧ (normop‘𝑇) ∈ ℝ ∧ ((normℎ‘𝐴) ∈ ℝ ∧ 0 < (normℎ‘𝐴))) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
65	47, 63, 21, 26, 64	syl112anc 1370	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
66	62, 65	mpbid 234	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
67	19, 66	pm2.61dane 3106	1 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   class class class wbr 5068  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   · cmul 10544   < clt 10677   ≤ cle 10678   / cdiv 11299  abscabs 14595   ℋchba 28698   ·_ℎ csm 28700  norm_ℎcno 28702  0_ℎc0v 28703  norm_opcnop 28724  ContOpccop 28725  LinOpclo 28726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-hilex 28778  ax-hfvadd 28779  ax-hvcom 28780  ax-hvass 28781  ax-hv0cl 28782  ax-hvaddid 28783  ax-hfvmul 28784  ax-hvmulid 28785  ax-hvmulass 28786  ax-hvdistr1 28787  ax-hvdistr2 28788  ax-hvmul0 28789  ax-hfi 28858  ax-his1 28861  ax-his2 28862  ax-his3 28863  ax-his4 28864

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-sup 8908  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-seq 13373  df-exp 13433  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-grpo 28272  df-gid 28273  df-ablo 28324  df-vc 28338  df-nv 28371  df-va 28374  df-ba 28375  df-sm 28376  df-0v 28377  df-nmcv 28379  df-hnorm 28747  df-hba 28748  df-hvsub 28750  df-nmop 29618  df-cnop 29619  df-lnop 29620

This theorem is referenced by:  nmcoplb  29809  cnlnadjlem2  29847  cnlnadjlem7  29852