Theorem nmopubt 9827

Description: An upper bound for an operator norm.

Assertion

Ref	Expression
nmopubt	⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) ≤ A)

Distinct variable groups:   x,A   x,T

Proof of Theorem nmopubt

Step	Hyp	Ref	Expression
1		nmopvalt 9777	. . 3 ⊢ (T: ℋ –→ ℋ → (normop ‘T) = sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
2	1	3ad2ant1 802	. 2 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) = sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ))
3		supxrleub 6101	. . 3 ⊢ (({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ* ⋀ A ∈ ℝ* ⋀ ∀w ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}w ≤ A) → sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ) ≤ A)
4		nmopsetretHIL 9786	. . . . 5 ⊢ (T: ℋ –→ ℋ → {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ)
5		ressxr 5510	. . . . . 6 ⊢ ℝ ⊆ ℝ*
6	5	a1i 8	. . . . 5 ⊢ (T: ℋ –→ ℋ → ℝ ⊆ ℝ*)
7	4, 6	sstrd 2077	. . . 4 ⊢ (T: ℋ –→ ℋ → {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ*)
8	7	3ad2ant1 802	. . 3 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ⊆ ℝ*)
9		3simp2 791	. . 3 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → A ∈ ℝ*)
10		fveq2 3730	. . . . . . . . . . . 12 ⊢ (x = z → (normh ‘x) = (normh ‘z))
11	10	breq1d 2634	. . . . . . . . . . 11 ⊢ (x = z → ((normh ‘x) ≤ 1 ↔ (normh ‘z) ≤ 1))
12		fveq2 3730	. . . . . . . . . . . . 13 ⊢ (x = z → (T ‘x) = (T ‘z))
13	12	fveq2d 3734	. . . . . . . . . . . 12 ⊢ (x = z → (normh ‘(T ‘x)) = (normh ‘(T ‘z)))
14	13	breq1d 2634	. . . . . . . . . . 11 ⊢ (x = z → ((normh ‘(T ‘x)) ≤ A ↔ (normh ‘(T ‘z)) ≤ A))
15	11, 14	imbi12d 628	. . . . . . . . . 10 ⊢ (x = z → (((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) ↔ ((normh ‘z) ≤ 1 → (normh ‘(T ‘z)) ≤ A)))
16	15	rcla4cv 1877	. . . . . . . . 9 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → (z ∈ ℋ → ((normh ‘z) ≤ 1 → (normh ‘(T ‘z)) ≤ A)))
17		breq1 2627	. . . . . . . . . 10 ⊢ (w = (normh ‘(T ‘z)) → (w ≤ A ↔ (normh ‘(T ‘z)) ≤ A))
18	17	biimprcd 156	. . . . . . . . 9 ⊢ ((normh ‘(T ‘z)) ≤ A → (w = (normh ‘(T ‘z)) → w ≤ A))
19	16, 18	syl8 24	. . . . . . . 8 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → (z ∈ ℋ → ((normh ‘z) ≤ 1 → (w = (normh ‘(T ‘z)) → w ≤ A))))
20	19	imp4a 364	. . . . . . 7 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → (z ∈ ℋ → (((normh ‘z) ≤ 1 ⋀ w = (normh ‘(T ‘z))) → w ≤ A)))
21	20	r19.23adv 1749	. . . . . 6 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → (∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ w = (normh ‘(T ‘z))) → w ≤ A))
22		visset 1816	. . . . . . 7 ⊢ w ∈ V
23		eqeq1 1484	. . . . . . . . 9 ⊢ (y = w → (y = (normh ‘(T ‘z)) ↔ w = (normh ‘(T ‘z))))
24	23	anbi2d 618	. . . . . . . 8 ⊢ (y = w → (((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z))) ↔ ((normh ‘z) ≤ 1 ⋀ w = (normh ‘(T ‘z)))))
25	24	rexbidv 1667	. . . . . . 7 ⊢ (y = w → (∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z))) ↔ ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ w = (normh ‘(T ‘z)))))
26	22, 25	elab 1900	. . . . . 6 ⊢ (w ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} ↔ ∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ w = (normh ‘(T ‘z))))
27	21, 26	syl5ib 206	. . . . 5 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → (w ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))} → w ≤ A))
28	27	r19.21aiv 1716	. . . 4 ⊢ (∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A) → ∀w ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}w ≤ A)
29	28	3ad2ant3 804	. . 3 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → ∀w ∈ {y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}w ≤ A)
30	3, 8, 9, 29	syl3anc 860	. 2 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → sup({y∣∃z ∈ ℋ ((normh ‘z) ≤ 1 ⋀ y = (normh ‘(T ‘z)))}, ℝ*, < ) ≤ A)
31	2, 30	eqbrtrd 2640	1 ⊢ ((T: ℋ –→ ℋ ⋀ A ∈ ℝ* ⋀ ∀x ∈ ℋ ((normh ‘x) ≤ 1 → (normh ‘(T ‘x)) ≤ A)) → (normop ‘T) ≤ A)

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  {cab 1466  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050   class class class wbr 2624  –→wf 3184   ‘cfv 3188  supcsup 4582  ℝcr 5245  1c1 5247   ≤ cle 5307  ℝ^*cxr 5497   < clt 5498   ℋ chil 8783  norm_hcno 8789  norm_opcnop 8809

This theorem is referenced by:  nmopub2tALT 9828  nmophm 9956  nmopadjlem 10017  nmoptri 10022  nmopco 10023  nmopcoadj 10029

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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