Theorem nmcoplbi 31964

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 12294	. . . . 5 ⊢ 0 ≤ 0
2	1	a1i 11	. . . 4 ⊢ (𝐴 = 0ℎ → 0 ≤ 0)
3		fveq2 6861	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
4		nmcopex.1	. . . . . . . 8 ⊢ 𝑇 ∈ LinOp
5	4	lnop0i 31906	. . . . . . 7 ⊢ (𝑇‘0ℎ) = 0ℎ
6	3, 5	eqtrdi 2781	. . . . . 6 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = 0ℎ)
7	6	fveq2d 6865	. . . . 5 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = (normℎ‘0ℎ))
8		norm0 31064	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
9	7, 8	eqtrdi 2781	. . . 4 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = 0)
10		fveq2 6861	. . . . . . 7 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
11	10, 8	eqtrdi 2781	. . . . . 6 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = 0)
12	11	oveq2d 7406	. . . . 5 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘𝑇) · 0))
13		nmcopex.2	. . . . . . . 8 ⊢ 𝑇 ∈ ContOp
14	4, 13	nmcopexi 31963	. . . . . . 7 ⊢ (normop‘𝑇) ∈ ℝ
15	14	recni 11195	. . . . . 6 ⊢ (normop‘𝑇) ∈ ℂ
16	15	mul01i 11371	. . . . 5 ⊢ ((normop‘𝑇) · 0) = 0
17	12, 16	eqtrdi 2781	. . . 4 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = 0)
18	2, 9, 17	3brtr4d 5142	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
19	18	adantl 481	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 = 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
20		normcl 31061	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
22		normne0 31066	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
23	22	biimpar 477	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
24	21, 23	rereccld 12016	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
25		normgt0 31063	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
26	25	biimpa 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
27	21, 26	recgt0d 12124	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
28		0re 11183	. . . . . . . . 9 ⊢ 0 ∈ ℝ
29		ltle 11269	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
30	28, 29	mpan 690	. . . . . . . 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 15396	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
33	32	oveq1d 7405	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
34	24	recnd 11209	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
35		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
36	4	lnopmuli 31908	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
37	34, 35, 36	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
38	37	fveq2d 6865	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))))
39	4	lnopfi 31905	. . . . . . . . 9 ⊢ 𝑇: ℋ⟶ ℋ
40	39	ffvelcdmi 7058	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
41	40	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℋ)
42		norm-iii 31076	. . . . . . 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 584	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
44	38, 43	eqtrd 2765	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
45		normcl 31061	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
46	40, 45	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
48	47	recnd 11209	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℂ)
49	21	recnd 11209	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
50	48, 49, 23	divrec2d 11969	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
51	33, 44, 50	3eqtr4rd 2776	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
52		hvmulcl 30949	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
53	34, 35, 52	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
54		normcl 31061	. . . . . . 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 31185	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
57		eqle 11283	. . . . . 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 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
59		nmoplb 31843	. . . . . 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 1450	. . . . 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 584	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
62	51, 61	eqbrtrd 5132	. . 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 12069	. . . 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 1376	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
66	62, 65	mpbid 232	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
67	19, 66	pm2.61dane 3013	1 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080   < clt 11215   ≤ cle 11216   / cdiv 11842  abscabs 15207   ℋchba 30855   ·_ℎ csm 30857  norm_ℎcno 30859  0_ℎc0v 30860  norm_opcnop 30881  ContOpccop 30882  LinOpclo 30883

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-hilex 30935  ax-hfvadd 30936  ax-hvcom 30937  ax-hvass 30938  ax-hv0cl 30939  ax-hvaddid 30940  ax-hfvmul 30941  ax-hvmulid 30942  ax-hvmulass 30943  ax-hvdistr1 30944  ax-hvdistr2 30945  ax-hvmul0 30946  ax-hfi 31015  ax-his1 31018  ax-his2 31019  ax-his3 31020  ax-his4 31021

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-seq 13974  df-exp 14034  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-grpo 30429  df-gid 30430  df-ablo 30481  df-vc 30495  df-nv 30528  df-va 30531  df-ba 30532  df-sm 30533  df-0v 30534  df-nmcv 30536  df-hnorm 30904  df-hba 30905  df-hvsub 30907  df-nmop 31775  df-cnop 31776  df-lnop 31777

This theorem is referenced by:  nmcoplb  31966  cnlnadjlem2  32004  cnlnadjlem7  32009