Theorem nmobndseqi 8436

Description: A bounded sequence determines a bounded operator.

Hypotheses

Ref	Expression
nmoubi.1	⊢ X = (Base ‘U)
nmoubi.y	⊢ Y = (Base ‘W)
nmoubi.l	⊢ L = (norm ‘U)
nmoubi.m	⊢ M = (norm ‘W)
nmoubi.3	⊢ N = (U normOp W)
nmoubi.u	⊢ U ∈ NrmCVec
nmoubi.w	⊢ W ∈ NrmCVec

Assertion

Ref	Expression
nmobndseqi	⊢ ((T:X–→Y ⋀ ∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k)) → (N ‘T) ∈ ℝ)

Distinct variable groups:   f,k,L   f,M,k   k,N   T,f,k   f,X,k   k,Y

Proof of Theorem nmobndseqi

Step	Hyp	Ref	Expression
1		nmoubi.1	. . . 4 ⊢ X = (Base ‘U)
2		nmoubi.y	. . . 4 ⊢ Y = (Base ‘W)
3		nmoubi.l	. . . 4 ⊢ L = (norm ‘U)
4		nmoubi.m	. . . 4 ⊢ M = (norm ‘W)
5		nmoubi.3	. . . 4 ⊢ N = (U normOp W)
6		nmoubi.u	. . . 4 ⊢ U ∈ NrmCVec
7		nmoubi.w	. . . 4 ⊢ W ∈ NrmCVec
8	1, 2, 3, 4, 5, 6, 7	nmobndi 8434	. . 3 ⊢ (T:X–→Y → ((N ‘T) ∈ ℝ ↔ ∃k ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k)))
9		impexp 347	. . . . . 6 ⊢ (((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k) ↔ (f:ℕ–→X → (∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1 → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k)))
10		r19.35 1762	. . . . . . 7 ⊢ (∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k) ↔ (∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1 → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k))
11	10	imbi2i 185	. . . . . 6 ⊢ ((f:ℕ–→X → ∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)) ↔ (f:ℕ–→X → (∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1 → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k)))
12	9, 11	bitr4 176	. . . . 5 ⊢ (((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k) ↔ (f:ℕ–→X → ∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)))
13	12	albii 1001	. . . 4 ⊢ (∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k) ↔ ∀f(f:ℕ–→X → ∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)))
14		nnex 5935	. . . . . 6 ⊢ ℕ ∈ V
15		fveq2 3730	. . . . . . . 8 ⊢ (y = (f ‘k) → (L ‘y) = (L ‘(f ‘k)))
16	15	breq1d 2634	. . . . . . 7 ⊢ (y = (f ‘k) → ((L ‘y) ≤ 1 ↔ (L ‘(f ‘k)) ≤ 1))
17		fveq2 3730	. . . . . . . . 9 ⊢ (y = (f ‘k) → (T ‘y) = (T ‘(f ‘k)))
18	17	fveq2d 3734	. . . . . . . 8 ⊢ (y = (f ‘k) → (M ‘(T ‘y)) = (M ‘(T ‘(f ‘k))))
19	18	breq1d 2634	. . . . . . 7 ⊢ (y = (f ‘k) → ((M ‘(T ‘y)) ≤ k ↔ (M ‘(T ‘(f ‘k))) ≤ k))
20	16, 19	imbi12d 628	. . . . . 6 ⊢ (y = (f ‘k) → (((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k) ↔ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)))
21	14, 20	ac6n 4767	. . . . 5 ⊢ (∀f(f:ℕ–→X → ∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)) → ∃k ∈ ℕ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k))
22		nnret 5931	. . . . . . 7 ⊢ (k ∈ ℕ → k ∈ ℝ)
23	22	anim1i 334	. . . . . 6 ⊢ ((k ∈ ℕ ⋀ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k)) → (k ∈ ℝ ⋀ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k)))
24	23	r19.22i2 1736	. . . . 5 ⊢ (∃k ∈ ℕ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k) → ∃k ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k))
25	21, 24	syl 10	. . . 4 ⊢ (∀f(f:ℕ–→X → ∃k ∈ ℕ ((L ‘(f ‘k)) ≤ 1 → (M ‘(T ‘(f ‘k))) ≤ k)) → ∃k ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k))
26	13, 25	sylbi 199	. . 3 ⊢ (∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k) → ∃k ∈ ℝ ∀y ∈ X ((L ‘y) ≤ 1 → (M ‘(T ‘y)) ≤ k))
27	8, 26	syl5bir 210	. 2 ⊢ (T:X–→Y → (∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k) → (N ‘T) ∈ ℝ))
28	27	imp 350	1 ⊢ ((T:X–→Y ⋀ ∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (L ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (M ‘(T ‘(f ‘k))) ≤ k)) → (N ‘T) ∈ ℝ)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   class class class wbr 2624  –→wf 3184   ‘cfv 3188  (class class class)co 3969  ℝcr 5245  1c1 5247   ≤ cle 5307  ℕcn 5308  NrmCVeccnv 8199  Basecba 8201  normcnm 8205   normOp cnmo 8398

This theorem is referenced by:  htthlem12 8627

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-reg 4602  ax-inf2 4634  ax-ac 4754

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-iin 2573  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-r1 4653  df-rank 4654  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-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-nmo 8402

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