Theorem htthlem12 8561

Description: Lemma for htthi 8562. Linear operator T is bounded.

Hypotheses

Ref	Expression
htthlem3.1	⊢ X = (Base ‘U)
htthlem3.p	⊢ P = ( ·i ‘U)
htthlem3.l	⊢ L = (U LnOp U)
htthlem3.b	⊢ B = (U BLnOp U)
htthlem3.u	⊢ U ∈ CHil
htthlem3.t	⊢ T ∈ L
htthlem3.a	⊢ ((x ∈ X ⋀ y ∈ X) → ((T ‘x)Py) = (xP(T ‘y)))
htthlem3.f	⊢ F = {⟨m, w⟩∣(m ∈ ℕ ⋀ w = {⟨v, u⟩∣(v ∈ X ⋀ u = ((T ‘v)P(f ‘m)))})}
htthlem3.c	⊢ C = ⟨⟨ + , · ⟩, abs⟩
htthlem3.d	⊢ D = (U BLnOp C)
htthlem3.n	⊢ N = (norm ‘U)
htthlem3.o	⊢ O = (U normOp C)

Assertion

Ref	Expression
htthlem12	⊢ T ∈ B

Distinct variable groups:   v,u,x,y   f,N   u,m,v,w,x,y,P   f,m,u,v,w,x,y,T   u,U,v   f,X,m,u,v,w,x,y

Proof of Theorem htthlem12

Step	Hyp	Ref	Expression
1		htthlem3.u	. . . 4 ⊢ U ∈ CHil
2	1	hlnvi 8527	. . 3 ⊢ U ∈ NrmCVec
3		eqid 1468	. . . 4 ⊢ (U normOp U) = (U normOp U)
4		htthlem3.l	. . . 4 ⊢ L = (U LnOp U)
5		htthlem3.b	. . . 4 ⊢ B = (U BLnOp U)
6	3, 4, 5	isblo2 8375	. . 3 ⊢ ((U ∈ NrmCVec ⋀ U ∈ NrmCVec) → (T ∈ B ↔ (T ∈ L ⋀ ((U normOp U) ‘T) ∈ ℝ)))
7	2, 2, 6	mp2an 695	. 2 ⊢ (T ∈ B ↔ (T ∈ L ⋀ ((U normOp U) ‘T) ∈ ℝ))
8		htthlem3.t	. 2 ⊢ T ∈ L
9		htthlem3.1	. . . . . 6 ⊢ X = (Base ‘U)
10	9, 9, 4	lnof 8350	. . . . 5 ⊢ ((U ∈ NrmCVec ⋀ U ∈ NrmCVec ⋀ T ∈ L) → T:X–→X)
11	2, 2, 8, 10	mp3an 913	. . . 4 ⊢ T:X–→X
12		htthlem3.p	. . . . . . . . 9 ⊢ P = ( ·i ‘U)
13		htthlem3.a	. . . . . . . . 9 ⊢ ((x ∈ X ⋀ y ∈ X) → ((T ‘x)Py) = (xP(T ‘y)))
14		htthlem3.f	. . . . . . . . 9 ⊢ F = {⟨m, w⟩∣(m ∈ ℕ ⋀ w = {⟨v, u⟩∣(v ∈ X ⋀ u = ((T ‘v)P(f ‘m)))})}
15		htthlem3.c	. . . . . . . . 9 ⊢ C = ⟨⟨ + , · ⟩, abs⟩
16		htthlem3.d	. . . . . . . . 9 ⊢ D = (U BLnOp C)
17		htthlem3.n	. . . . . . . . 9 ⊢ N = (norm ‘U)
18		htthlem3.o	. . . . . . . . 9 ⊢ O = (U normOp C)
19	9, 12, 4, 5, 1, 8, 13, 14, 15, 16, 17, 18	htthlem11 8560	. . . . . . . 8 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃z ∈ ℝ ∀k ∈ ℕ (O ‘(F ‘k)) ≤ z)
20	9, 12, 4, 5, 1, 8, 13, 14, 15, 16, 17, 18	htthlem10 8559	. . . . . . . . . . . . . 14 ⊢ (((f:ℕ–→X ⋀ k ∈ ℕ) ⋀ (z ∈ ℝ ⋀ (O ‘(F ‘k)) ≤ z)) → (N ‘(T ‘(f ‘k))) ≤ z)
21	20	exp32 377	. . . . . . . . . . . . 13 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (z ∈ ℝ → ((O ‘(F ‘k)) ≤ z → (N ‘(T ‘(f ‘k))) ≤ z)))
22	21	imp 350	. . . . . . . . . . . 12 ⊢ (((f:ℕ–→X ⋀ k ∈ ℕ) ⋀ z ∈ ℝ) → ((O ‘(F ‘k)) ≤ z → (N ‘(T ‘(f ‘k))) ≤ z))
23	22	an1rs 488	. . . . . . . . . . 11 ⊢ (((f:ℕ–→X ⋀ z ∈ ℝ) ⋀ k ∈ ℕ) → ((O ‘(F ‘k)) ≤ z → (N ‘(T ‘(f ‘k))) ≤ z))
24	23	r19.20dva 1701	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ z ∈ ℝ) → (∀k ∈ ℕ (O ‘(F ‘k)) ≤ z → ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z))
25	24	r19.22dva 1731	. . . . . . . . 9 ⊢ (f:ℕ–→X → (∃z ∈ ℝ ∀k ∈ ℕ (O ‘(F ‘k)) ≤ z → ∃z ∈ ℝ ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z))
26	25	adantr 389	. . . . . . . 8 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → (∃z ∈ ℝ ∀k ∈ ℕ (O ‘(F ‘k)) ≤ z → ∃z ∈ ℝ ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z))
27	19, 26	mpd 26	. . . . . . 7 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃z ∈ ℝ ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z)
28	9, 4, 1, 8	htthlem1 8550	. . . . . . . . . . . . 13 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (T ‘(f ‘k)) ∈ X)
29	9, 17	nvcl 8227	. . . . . . . . . . . . . 14 ⊢ ((U ∈ NrmCVec ⋀ (T ‘(f ‘k)) ∈ X) → (N ‘(T ‘(f ‘k))) ∈ ℝ)
30	2, 29	mpan 693	. . . . . . . . . . . . 13 ⊢ ((T ‘(f ‘k)) ∈ X → (N ‘(T ‘(f ‘k))) ∈ ℝ)
31	28, 30	syl 10	. . . . . . . . . . . 12 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → (N ‘(T ‘(f ‘k))) ∈ ℝ)
32	31	anim1i 334	. . . . . . . . . . 11 ⊢ (((f:ℕ–→X ⋀ k ∈ ℕ) ⋀ (N ‘(T ‘(f ‘k))) ≤ z) → ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z))
33	32	ex 373	. . . . . . . . . 10 ⊢ ((f:ℕ–→X ⋀ k ∈ ℕ) → ((N ‘(T ‘(f ‘k))) ≤ z → ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z)))
34	33	r19.20dva 1701	. . . . . . . . 9 ⊢ (f:ℕ–→X → (∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z → ∀k ∈ ℕ ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z)))
35	34	r19.22sdv 1730	. . . . . . . 8 ⊢ (f:ℕ–→X → (∃z ∈ ℝ ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z → ∃z ∈ ℝ ∀k ∈ ℕ ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z)))
36	35	adantr 389	. . . . . . 7 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → (∃z ∈ ℝ ∀k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ z → ∃z ∈ ℝ ∀k ∈ ℕ ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z)))
37	27, 36	mpd 26	. . . . . 6 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃z ∈ ℝ ∀k ∈ ℕ ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z))
38		bndndx 6020	. . . . . 6 ⊢ (∃z ∈ ℝ ∀k ∈ ℕ ((N ‘(T ‘(f ‘k))) ∈ ℝ ⋀ (N ‘(T ‘(f ‘k))) ≤ z) → ∃k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ k)
39	37, 38	syl 10	. . . . 5 ⊢ ((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ k)
40	39	ax-gen 960	. . . 4 ⊢ ∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ k)
41	11, 40	pm3.2i 285	. . 3 ⊢ (T:X–→X ⋀ ∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ k))
42	9, 9, 17, 17, 3, 2, 2	nmobndseqi 8372	. . 3 ⊢ ((T:X–→X ⋀ ∀f((f:ℕ–→X ⋀ ∀k ∈ ℕ (N ‘(f ‘k)) ≤ 1) → ∃k ∈ ℕ (N ‘(T ‘(f ‘k))) ≤ k)) → ((U normOp U) ‘T) ∈ ℝ)
43	41, 42	ax-mp 7	. 2 ⊢ ((U normOp U) ‘T) ∈ ℝ
44	7, 8, 43	mpbir2an 728	1 ⊢ T ∈ B

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 951   = wceq 953   ∈ wcel 955  ∀wral 1637  ∃wrex 1638  ⟨cop 2401   class class class wbr 2609  {copab 2656  –→wf 3168   ‘cfv 3172  (class class class)co 3948  ℝcr 5205  1c1 5207   + caddc 5209   · cmul 5211   ≤ cle 5267  ℕcn 5268  abscabs 6681  NrmCVeccnv 8141  Basecba 8143  normcnm 8147   ·_i cip 8283   LnOp clno 8335   normOp cnmo 8336   BLnOp cblo 8337  CHilchl 8520

This theorem is referenced by:  htthi 8562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-iso 3189  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-r1 4615  df-rank 4616  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-fl 6172  df-q 6194  df-seq1 6245  df-shft 6278  df-ioo 6298  df-uz 6350  df-fz 6400  df-seqz 6465  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913  df-sum 6918  df-top 7534  df-bases 7536  df-topgen 7537  df-cld 7605  df-ntr 7606  df-cls 7607  df-nei 7654  df-lp 7682  df-cn 7694  df-cnp 7695  df-haus 7721  df-met 7732  df-bl 7734  df-opn 7735  df-lm 7860  df-cau 7861  df-cmet 7862  df-grp 7971  df-gid 7972  df-ginv 7973  df-gdiv 7974  df-abl 8036  df-vc 8102  df-nv 8149  df-va 8152  df-ba 8153  df-sm 8154  df-0v 8155  df-vs 8156  df-nm 8157  df-ims 8158  df-ip 8284  df-lno 8339  df-nmo 8340  df-blo 8341  df-0o 8342  df-ph 8403  df-bn 8454  df-hl 8521

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