Theorem nmopub2tALT 9828

Description: An upper bound for an operator norm.

Assertion

Ref	Expression
nmopub2tALT	⊢ ((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A) ⋀ ∀x ∈ ℋ (normh ‘(T ‘x)) ≤ (A · (normh ‘x))) → (normop ‘T) ≤ A)

Distinct variable groups:   x,A   x,T

Proof of Theorem nmopub2tALT

Step	Hyp	Ref	Expression
1		1re 5447	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
2		lemul2itOLD 5842	. . . . . . . . . . 11 ⊢ ((((normh ‘x) ∈ ℝ ⋀ 1 ∈ ℝ ⋀ A ∈ ℝ) ⋀ (0 ≤ A ⋀ (normh ‘x) ≤ 1)) → (A · (normh ‘x)) ≤ (A · 1))
3	1, 2	mp3anl2 913	. . . . . . . . . 10 ⊢ ((((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ) ⋀ (0 ≤ A ⋀ (normh ‘x) ≤ 1)) → (A · (normh ‘x)) ≤ (A · 1))
4		normclt 8986	. . . . . . . . . . . . . . 15 ⊢ (x ∈ ℋ → (normh ‘x) ∈ ℝ)
5	4	anim1i 334	. . . . . . . . . . . . . 14 ⊢ ((x ∈ ℋ ⋀ A ∈ ℝ) → ((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ))
6	5	ancoms 438	. . . . . . . . . . . . 13 ⊢ ((A ∈ ℝ ⋀ x ∈ ℋ ) → ((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ))
7	6	adantlr 395	. . . . . . . . . . . 12 ⊢ (((A ∈ ℝ ⋀ 0 ≤ A) ⋀ x ∈ ℋ ) → ((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ))
8	7	adantll 394	. . . . . . . . . . 11 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → ((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ))
9	8	adantr 391	. . . . . . . . . 10 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → ((normh ‘x) ∈ ℝ ⋀ A ∈ ℝ))
10		id 59	. . . . . . . . . . . . 13 ⊢ ((0 ≤ A ⋀ (normh ‘x) ≤ 1) → (0 ≤ A ⋀ (normh ‘x) ≤ 1))
11	10	adantll 394	. . . . . . . . . . . 12 ⊢ (((A ∈ ℝ ⋀ 0 ≤ A) ⋀ (normh ‘x) ≤ 1) → (0 ≤ A ⋀ (normh ‘x) ≤ 1))
12	11	adantll 394	. . . . . . . . . . 11 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ (normh ‘x) ≤ 1) → (0 ≤ A ⋀ (normh ‘x) ≤ 1))
13	12	adantlr 395	. . . . . . . . . 10 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → (0 ≤ A ⋀ (normh ‘x) ≤ 1))
14	3, 9, 13	sylanc 473	. . . . . . . . 9 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → (A · (normh ‘x)) ≤ (A · 1))
15		recnt 5325	. . . . . . . . . . . 12 ⊢ (A ∈ ℝ → A ∈ ℂ)
16		ax1id 5294	. . . . . . . . . . . 12 ⊢ (A ∈ ℂ → (A · 1) = A)
17	15, 16	syl 10	. . . . . . . . . . 11 ⊢ (A ∈ ℝ → (A · 1) = A)
18	17	ad2antrl 408	. . . . . . . . . 10 ⊢ ((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) → (A · 1) = A)
19	18	ad2antrr 406	. . . . . . . . 9 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → (A · 1) = A)
20	14, 19	breqtrd 2644	. . . . . . . 8 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → (A · (normh ‘x)) ≤ A)
21		letrt 5537	. . . . . . . . . 10 ⊢ (((normh ‘(T ‘x)) ∈ ℝ ⋀ (A · (normh ‘x)) ∈ ℝ ⋀ A ∈ ℝ) → (((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) ⋀ (A · (normh ‘x)) ≤ A) → (normh ‘(T ‘x)) ≤ A))
22		ffvelrn 3820	. . . . . . . . . . . 12 ⊢ ((T: ℋ –→ ℋ ⋀ x ∈ ℋ ) → (T ‘x) ∈ ℋ )
23		normclt 8986	. . . . . . . . . . . 12 ⊢ ((T ‘x) ∈ ℋ → (normh ‘(T ‘x)) ∈ ℝ)
24	22, 23	syl 10	. . . . . . . . . . 11 ⊢ ((T: ℋ –→ ℋ ⋀ x ∈ ℋ ) → (normh ‘(T ‘x)) ∈ ℝ)
25	24	adantlr 395	. . . . . . . . . 10 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → (normh ‘(T ‘x)) ∈ ℝ)
26		axmulrcl 5286	. . . . . . . . . . . . 13 ⊢ ((A ∈ ℝ ⋀ (normh ‘x) ∈ ℝ) → (A · (normh ‘x)) ∈ ℝ)
27	26, 4	sylan2 453	. . . . . . . . . . . 12 ⊢ ((A ∈ ℝ ⋀ x ∈ ℋ ) → (A · (normh ‘x)) ∈ ℝ)
28	27	adantlr 395	. . . . . . . . . . 11 ⊢ (((A ∈ ℝ ⋀ 0 ≤ A) ⋀ x ∈ ℋ ) → (A · (normh ‘x)) ∈ ℝ)
29	28	adantll 394	. . . . . . . . . 10 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → (A · (normh ‘x)) ∈ ℝ)
30		pm3.26 319	. . . . . . . . . . 11 ⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → A ∈ ℝ)
31	30	ad2antlr 407	. . . . . . . . . 10 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → A ∈ ℝ)
32	21, 25, 29, 31	syl3anc 860	. . . . . . . . 9 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → (((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) ⋀ (A · (normh ‘x)) ≤ A) → (normh ‘(T ‘x)) ≤ A))
33	32	adantr 391	. . . . . . . 8 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → (((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) ⋀ (A · (normh ‘x)) ≤ A) → (normh ‘(T ‘x)) ≤ A))
34	20, 33	mpan2d 704	. . . . . . 7 ⊢ ((((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) ⋀ (normh ‘x) ≤ 1) → ((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) → (normh ‘(T ‘x)) ≤ A))
35	34	ex 373	. . . . . 6 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → ((normh ‘x) ≤ 1 → ((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) → (normh ‘(T ‘x)) ≤ A)))
36	35	com23 32	. . . . 5 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ x ∈ ℋ ) → ((normh ‘(T ‘x)) ≤ (A · (normh ‘x)) → ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)))
37	36	r19.20dva 1712	. . . 4 ⊢ ((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) → (∀x ∈ ℋ (normh ‘(T ‘x)) ≤ (A · (normh ‘x)) → ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)))
38	37	imp 350	. . 3 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ ∀x ∈ ℋ (normh ‘(T ‘x)) ≤ (A · (normh ‘x))) → ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A))
39		nmopubt 9827	. . . . 5 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) ≤ A)
40		rexrt 5511	. . . . . 6 ⊢ (A ∈ ℝ → A ∈ ℝ*)
41	40	adantr 391	. . . . 5 ⊢ ((A ∈ ℝ ⋀ 0 ≤ A) → A ∈ ℝ*)
42	39, 41	syl3an2 862	. . . 4 ⊢ ((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A) ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) ≤ A)
43	42	3expa 835	. . 3 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) ≤ A)
44	38, 43	syldan 469	. 2 ⊢ (((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A)) ⋀ ∀x ∈ ℋ (normh ‘(T ‘x)) ≤ (A · (normh ‘x))) → (normop ‘T) ≤ A)
45	44	3impa 830	1 ⊢ ((T: ℋ –→ ℋ ⋀ (A ∈ ℝ ⋀ 0 ≤ A) ⋀ ∀x ∈ ℋ (normh ‘(T ‘x)) ≤ (A · (normh ‘x))) → (normop ‘T) ≤ A)

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∀wral 1648   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  ℂcc 5244  ℝcr 5245  0cc0 5246  1c1 5247   · cmul 5251   ≤ cle 5307  ℝ^*cxr 5497   ℋ chil 8783  norm_hcno 8789  norm_opcnop 8809

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-grp 8034  df-gid 8035  df-ginv 8036  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215  df-hnorm 8832  df-hvsub 8835  df-nmop 9760

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