Theorem nmbdoplbi 30966

Description: A lower bound for the norm of a bounded linear operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nmbdoplb.1	⊢ 𝑇 ∈ BndLinOp

Assertion

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

Proof of Theorem nmbdoplbi

Step	Hyp	Ref	Expression
1		fveq2 6842	. . . 4 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2	1	fveq2d 6846	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘0ℎ)))
3		fveq2 6842	. . . 4 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
4	3	oveq2d 7373	. . 3 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘𝑇) · (normℎ‘0ℎ)))
5	2, 4	breq12d 5118	. 2 ⊢ (𝐴 = 0ℎ → ((normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)) ↔ (normℎ‘(𝑇‘0ℎ)) ≤ ((normop‘𝑇) · (normℎ‘0ℎ))))
6		nmbdoplb.1	. . . . . . . . . . . 12 ⊢ 𝑇 ∈ BndLinOp
7		bdopln 30803	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp
9	8	lnopfi 30911	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
10	9	ffvelcdmi 7034	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
11		normcl 30067	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
14	13	recnd 11183	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℂ)
15		normcl 30067	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
16	15	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
17	16	recnd 11183	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
18		normne0 30072	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
19	18	biimpar 478	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
20	14, 17, 19	divrec2d 11935	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
21	16, 19	rereccld 11982	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
22	21	recnd 11183	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
23		simpl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
24	8	lnopmuli 30914	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
25	22, 23, 24	syl2anc 584	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
26	25	fveq2d 6846	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))))
27	10	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℋ)
28		norm-iii 30082	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ (𝑇‘𝐴) ∈ ℋ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
29	22, 27, 28	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
30		normgt0 30069	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
31	30	biimpa 477	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
32	16, 31	recgt0d 12089	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
33		0re 11157	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
34		ltle 11243	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
35	33, 34	mpan 688	. . . . . . . . 9 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
36	21, 32, 35	sylc 65	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
37	21, 36	absidd 15307	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
38	37	oveq1d 7372	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
39	26, 29, 38	3eqtrrd 2781	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
40	20, 39	eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
41		hvmulcl 29955	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
42	22, 23, 41	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
43		normcl 30067	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
44	42, 43	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
45		norm1 30191	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
46		eqle 11257	. . . . . 6 ⊢ (((normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
47	44, 45, 46	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
48		nmoplb 30849	. . . . . 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))
49	9, 48	mp3an1 1448	. . . . 5 ⊢ ((((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ ∧ (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
50	42, 47, 49	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) ≤ (normop‘𝑇))
51	40, 50	eqbrtrd 5127	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇))
52		nmopre 30812	. . . . . 6 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ)
53	6, 52	ax-mp 5	. . . . 5 ⊢ (normop‘𝑇) ∈ ℝ
54	53	a1i 11	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normop‘𝑇) ∈ ℝ)
55		ledivmul2 12034	. . . 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))))
56	13, 54, 16, 31, 55	syl112anc 1374	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
57	51, 56	mpbid 231	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
58		0le0 12254	. . . 4 ⊢ 0 ≤ 0
59	8	lnop0i 30912	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0ℎ
60	59	fveq2i 6845	. . . . 5 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ)
61		norm0 30070	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
62	60, 61	eqtri 2764	. . . 4 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0
63	61	oveq2i 7368	. . . . 5 ⊢ ((normop‘𝑇) · (normℎ‘0ℎ)) = ((normop‘𝑇) · 0)
64	53	recni 11169	. . . . . 6 ⊢ (normop‘𝑇) ∈ ℂ
65	64	mul01i 11345	. . . . 5 ⊢ ((normop‘𝑇) · 0) = 0
66	63, 65	eqtri 2764	. . . 4 ⊢ ((normop‘𝑇) · (normℎ‘0ℎ)) = 0
67	58, 62, 66	3brtr4i 5135	. . 3 ⊢ (normℎ‘(𝑇‘0ℎ)) ≤ ((normop‘𝑇) · (normℎ‘0ℎ))
68	67	a1i 11	. 2 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘0ℎ)) ≤ ((normop‘𝑇) · (normℎ‘0ℎ)))
69	5, 57, 68	pm2.61ne 3030	1 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   · cmul 11056   < clt 11189   ≤ cle 11190   / cdiv 11812  abscabs 15119   ℋchba 29861   ·_ℎ csm 29863  norm_ℎcno 29865  0_ℎc0v 29866  norm_opcnop 29887  LinOpclo 29889  BndLinOpcbo 29890

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-hilex 29941  ax-hfvadd 29942  ax-hvcom 29943  ax-hvass 29944  ax-hv0cl 29945  ax-hvaddid 29946  ax-hfvmul 29947  ax-hvmulid 29948  ax-hvmulass 29949  ax-hvdistr1 29950  ax-hvdistr2 29951  ax-hvmul0 29952  ax-hfi 30021  ax-his1 30024  ax-his2 30025  ax-his3 30026  ax-his4 30027

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-grpo 29435  df-gid 29436  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-nmcv 29542  df-hnorm 29910  df-hba 29911  df-hvsub 29913  df-nmop 30781  df-lnop 30783  df-bdop 30784

This theorem is referenced by:  nmbdoplb  30967  nmopcoadji  31043