Theorem nmbdoplbi 32506

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 6879	. . . 4 ⊢ (𝐴 = 0ℎ → (𝑇‘𝐴) = (𝑇‘0ℎ))
2	1	fveq2d 6883	. . 3 ⊢ (𝐴 = 0ℎ → (normℎ‘(𝑇‘𝐴)) = (normℎ‘(𝑇‘0ℎ)))
3		fveq2 6879	. . . 4 ⊢ (𝐴 = 0ℎ → (normℎ‘𝐴) = (normℎ‘0ℎ))
4	3	oveq2d 7430	. . 3 ⊢ (𝐴 = 0ℎ → ((normop‘𝑇) · (normℎ‘𝐴)) = ((normop‘𝑇) · (normℎ‘0ℎ)))
5	2, 4	breq12d 5116	. 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 32343	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)
8	6, 7	ax-mp 5	. . . . . . . . . . 11 ⊢ 𝑇 ∈ LinOp
9	8	lnopfi 32451	. . . . . . . . . 10 ⊢ 𝑇: ℋ⟶ ℋ
10	9	ffvelcdmi 7077	. . . . . . . . 9 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ)
11		normcl 31607	. . . . . . . . 9 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
13	12	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ)
14	13	recnd 11262	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ∈ ℂ)
15		normcl 31607	. . . . . . . 8 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ)
16	15	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℝ)
17	16	recnd 11262	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ∈ ℂ)
18		normne0 31612	. . . . . . 7 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0ℎ))
19	18	biimpar 483	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘𝐴) ≠ 0)
20	14, 17, 19	divrec2d 12020	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
21	16, 19	rereccld 12067	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℝ)
22	21	recnd 11262	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (1 / (normℎ‘𝐴)) ∈ ℂ)
23		simpl 488	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 𝐴 ∈ ℋ)
24	8	lnopmuli 32454	. . . . . . . 8 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
25	22, 23, 24	syl2anc 596	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = ((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴)))
26	25	fveq2d 6883	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))) = (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))))
27	10	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (𝑇‘𝐴) ∈ ℋ)
28		norm-iii 31622	. . . . . . 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 596	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ (𝑇‘𝐴))) = ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))))
30		normgt0 31609	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ ↔ 0 < (normℎ‘𝐴)))
31	30	biimpa 482	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (normℎ‘𝐴))
32	16, 31	recgt0d 12174	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (1 / (normℎ‘𝐴)))
33		0re 11235	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
34		ltle 11323	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (1 / (normℎ‘𝐴)) ∈ ℝ) → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
35	33, 34	mpan 703	. . . . . . . . 9 ⊢ ((1 / (normℎ‘𝐴)) ∈ ℝ → (0 < (1 / (normℎ‘𝐴)) → 0 ≤ (1 / (normℎ‘𝐴))))
36	21, 32, 35	sylc 66	. . . . . . . 8 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 ≤ (1 / (normℎ‘𝐴)))
37	21, 36	absidd 15511	. . . . . . 7 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (abs‘(1 / (normℎ‘𝐴))) = (1 / (normℎ‘𝐴)))
38	37	oveq1d 7429	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((abs‘(1 / (normℎ‘𝐴))) · (normℎ‘(𝑇‘𝐴))) = ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))))
39	26, 29, 38	3eqtrrd 2800	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) · (normℎ‘(𝑇‘𝐴))) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
40	20, 39	eqtrd 2795	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) = (normℎ‘(𝑇‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴))))
41		hvmulcl 31495	. . . . . 6 ⊢ (((1 / (normℎ‘𝐴)) ∈ ℂ ∧ 𝐴 ∈ ℋ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
42	22, 23, 41	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → ((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ)
43		normcl 31607	. . . . . . 7 ⊢ (((1 / (normℎ‘𝐴)) ·ℎ 𝐴) ∈ ℋ → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
44	42, 43	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ∈ ℝ)
45		norm1 31731	. . . . . 6 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1)
46		eqle 11337	. . . . . 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 596	. . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) ≤ 1)
48		nmoplb 32389	. . . . . 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 1477	. . . . 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 596	. . . 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 32352	. . . . . 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 12119	. . . 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 1401	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (((normℎ‘(𝑇‘𝐴)) / (normℎ‘𝐴)) ≤ (normop‘𝑇) ↔ (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴))))
57	51, 56	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))
58		0le0 12367	. . . 4 ⊢ 0 ≤ 0
59	8	lnop0i 32452	. . . . . 6 ⊢ (𝑇‘0ℎ) = 0ℎ
60	59	fveq2i 6882	. . . . 5 ⊢ (normℎ‘(𝑇‘0ℎ)) = (normℎ‘0ℎ)
61		norm0 31610	. . . . 5 ⊢ (normℎ‘0ℎ) = 0
62	60, 61	eqtri 2783	. . . 4 ⊢ (normℎ‘(𝑇‘0ℎ)) = 0
63	61	oveq2i 7425	. . . . 5 ⊢ ((normop‘𝑇) · (normℎ‘0ℎ)) = ((normop‘𝑇) · 0)
64	53	recni 11248	. . . . . 6 ⊢ (normop‘𝑇) ∈ ℂ
65	64	mul01i 11425	. . . . 5 ⊢ ((normop‘𝑇) · 0) = 0
66	63, 65	eqtri 2783	. . . 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 3040	1 ⊢ (𝐴 ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ≤ ((normop‘𝑇) · (normℎ‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  ℂcc 11123  ℝcr 11124  0cc0 11125  1c1 11126   · cmul 11130   < clt 11268   ≤ cle 11269   / cdiv 11896  abscabs 15322   ℋchba 31401   ·_ℎ csm 31403  norm_ℎcno 31405  0_ℎc0v 31406  norm_opcnop 31427  LinOpclo 31429  BndLinOpcbo 31430

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202  ax-pre-sup 11203  ax-hilex 31481  ax-hfvadd 31482  ax-hvcom 31483  ax-hvass 31484  ax-hv0cl 31485  ax-hvaddid 31486  ax-hfvmul 31487  ax-hvmulid 31488  ax-hvmulass 31489  ax-hvdistr1 31490  ax-hvdistr2 31491  ax-hvmul0 31492  ax-hfi 31561  ax-his1 31564  ax-his2 31565  ax-his3 31566  ax-his4 31567

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-sup 9413  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-div 11897  df-nn 12259  df-2 12328  df-3 12329  df-4 12330  df-n0 12530  df-z 12617  df-uz 12889  df-rp 13044  df-seq 14067  df-exp 14127  df-cj 15187  df-re 15188  df-im 15189  df-sqrt 15323  df-abs 15324  df-grpo 30975  df-gid 30976  df-ablo 31027  df-vc 31041  df-nv 31074  df-va 31077  df-ba 31078  df-sm 31079  df-0v 31080  df-nmcv 31082  df-hnorm 31450  df-hba 31451  df-hvsub 31453  df-nmop 32321  df-lnop 32323  df-bdop 32324

This theorem is used by:  nmbdoplb  32507  nmopcoadji  32583