Theorem hlimcauii 10739

Description: If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence.

Hypotheses

Ref	Expression
hlimcau.1	|- A e. _V
hlimcau.2	|- F e. _V
hlimcaui.4	|- F ~~>v A

Assertion

Ref	Expression
hlimcauii	|- F e. Cauchy

Proof of Theorem hlimcauii

Step	Hyp	Ref	Expression
1		hcau 10684	. 2 |- (F e. Cauchy <-> (F:NN-->~H /\ A.x e. RR (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x))))
2		hlimcaui.4	. . 3 |- F ~~>v A
3		hlimcau.2	. . . 4 |- F e. _V
4		hlimcau.1	. . . 4 |- A e. _V
5	3, 4	hlimseqi 10690	. . 3 |- (F ~~>v A -> F:NN-->~H)
6	2, 5	ax-mp 7	. 2 |- F:NN-->~H
7		rehalfcl 7220	. . . . . . 7 |- (x e. RR -> (x / 2) e. RR)
8	7	adantr 425	. . . . . 6 |- ((x e. RR /\ 0 < x) -> (x / 2) e. RR)
9		breq2 3342	. . . . . . . . 9 |- (x = if(x e. RR, x, 0) -> (0 < x <-> 0 < if(x e. RR, x, 0)))
10		opreq1 4889	. . . . . . . . . 10 |- (x = if(x e. RR, x, 0) -> (x / 2) = (if(x e. RR, x, 0) / 2))
11	10	breq2d 3350	. . . . . . . . 9 |- (x = if(x e. RR, x, 0) -> (0 < (x / 2) <-> 0 < (if(x e. RR, x, 0) / 2)))
12	9, 11	imbi12d 688	. . . . . . . 8 |- (x = if(x e. RR, x, 0) -> ((0 < x -> 0 < (x / 2)) <-> (0 < if(x e. RR, x, 0) -> 0 < (if(x e. RR, x, 0) / 2))))
13		0re 6603	. . . . . . . . . 10 |- 0 e. RR
14	13	elimel 3025	. . . . . . . . 9 |- if(x e. RR, x, 0) e. RR
15		2re 7163	. . . . . . . . 9 |- 2 e. RR
16		2pos 7173	. . . . . . . . 9 |- 0 < 2
17	14, 15, 16	divgt0i2i 7041	. . . . . . . 8 |- (0 < if(x e. RR, x, 0) -> 0 < (if(x e. RR, x, 0) / 2))
18	12, 17	dedth 3011	. . . . . . 7 |- (x e. RR -> (0 < x -> 0 < (x / 2)))
19	18	imp 377	. . . . . 6 |- ((x e. RR /\ 0 < x) -> 0 < (x / 2))
20		breq2 3342	. . . . . . . 8 |- (v = (x / 2) -> (0 < v <-> 0 < (x / 2)))
21		breq2 3342	. . . . . . . . . 10 |- (v = (x / 2) -> ((normh` ((F` z) -h A)) < v <-> (normh` ((F` z) -h A)) < (x / 2)))
22	21	imbi2d 674	. . . . . . . . 9 |- (v = (x / 2) -> ((y <_ z -> (normh` ((F` z) -h A)) < v) <-> (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
23	22	rexralbidv 2142	. . . . . . . 8 |- (v = (x / 2) -> (E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v) <-> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
24	20, 23	imbi12d 688	. . . . . . 7 |- (v = (x / 2) -> ((0 < v -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v)) <-> (0 < (x / 2) -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)))))
25	3, 4	hlimveci 10691	. . . . . . . . 9 |- (F ~~>v A -> A e. ~H)
26	2, 25	ax-mp 7	. . . . . . . 8 |- A e. ~H
27		hlim2 10693	. . . . . . . . 9 |- ((F:NN-->~H /\ A e. ~H) -> (F ~~>v A <-> A.v e. RR (0 < v -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v))))
28	2, 27	mpbii 210	. . . . . . . 8 |- ((F:NN-->~H /\ A e. ~H) -> A.v e. RR (0 < v -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v)))
29	6, 26, 28	mp2an 761	. . . . . . 7 |- A.v e. RR (0 < v -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v))
30	24, 29	vtoclri 2360	. . . . . 6 |- ((x / 2) e. RR -> (0 < (x / 2) -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
31	8, 19, 30	sylc 83	. . . . 5 |- ((x e. RR /\ 0 < x) -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)))
32		prth 615	. . . . . . . . . . 11 |- (((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) -> ((y <_ z /\ y <_ w) -> ((normh` ((F` z) -h A)) < (x / 2) /\ (normh` ((F` w) -h A)) < (x / 2))))
33		normsub 10643	. . . . . . . . . . . . . . . 16 |- ((A e. ~H /\ (F` w) e. ~H) -> (normh` (A -h (F` w))) = (normh` ((F` w) -h A)))
34	33	breq1d 3348	. . . . . . . . . . . . . . 15 |- ((A e. ~H /\ (F` w) e. ~H) -> ((normh` (A -h (F` w))) < (x / 2) <-> (normh` ((F` w) -h A)) < (x / 2)))
35	34	anbi2d 678	. . . . . . . . . . . . . 14 |- ((A e. ~H /\ (F` w) e. ~H) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` (A -h (F` w))) < (x / 2)) <-> ((normh` ((F` z) -h A)) < (x / 2) /\ (normh` ((F` w) -h A)) < (x / 2))))
36	35	adantl 424	. . . . . . . . . . . . 13 |- (((x e. RR /\ z e. NN) /\ (A e. ~H /\ (F` w) e. ~H)) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` (A -h (F` w))) < (x / 2)) <-> ((normh` ((F` z) -h A)) < (x / 2) /\ (normh` ((F` w) -h A)) < (x / 2))))
37	6	ffvelrni 4788	. . . . . . . . . . . . . . . . . . . 20 |- (z e. NN -> (F` z) e. ~H)
38	37	anim1i 361	. . . . . . . . . . . . . . . . . . 19 |- ((z e. NN /\ (F` w) e. ~H) -> ((F` z) e. ~H /\ (F` w) e. ~H))
39	38	ancoms 484	. . . . . . . . . . . . . . . . . 18 |- (((F` w) e. ~H /\ z e. NN) -> ((F` z) e. ~H /\ (F` w) e. ~H))
40	39	anim1i 361	. . . . . . . . . . . . . . . . 17 |- ((((F` w) e. ~H /\ z e. NN) /\ (A e. ~H /\ x e. RR)) -> (((F` z) e. ~H /\ (F` w) e. ~H) /\ (A e. ~H /\ x e. RR)))
41	40	ancoms 484	. . . . . . . . . . . . . . . 16 |- (((A e. ~H /\ x e. RR) /\ ((F` w) e. ~H /\ z e. NN)) -> (((F` z) e. ~H /\ (F` w) e. ~H) /\ (A e. ~H /\ x e. RR)))
42	41	an4s 566	. . . . . . . . . . . . . . 15 |- (((A e. ~H /\ (F` w) e. ~H) /\ (x e. RR /\ z e. NN)) -> (((F` z) e. ~H /\ (F` w) e. ~H) /\ (A e. ~H /\ x e. RR)))
43	42	ancoms 484	. . . . . . . . . . . . . 14 |- (((x e. RR /\ z e. NN) /\ (A e. ~H /\ (F` w) e. ~H)) -> (((F` z) e. ~H /\ (F` w) e. ~H) /\ (A e. ~H /\ x e. RR)))
44		norm3lemt 10652	. . . . . . . . . . . . . 14 |- ((((F` z) e. ~H /\ (F` w) e. ~H) /\ (A e. ~H /\ x e. RR)) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` (A -h (F` w))) < (x / 2)) -> (normh` ((F` z) -h (F` w))) < x))
45	43, 44	syl 12	. . . . . . . . . . . . 13 |- (((x e. RR /\ z e. NN) /\ (A e. ~H /\ (F` w) e. ~H)) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` (A -h (F` w))) < (x / 2)) -> (normh` ((F` z) -h (F` w))) < x))
46	36, 45	sylbird 222	. . . . . . . . . . . 12 |- (((x e. RR /\ z e. NN) /\ (A e. ~H /\ (F` w) e. ~H)) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` ((F` w) -h A)) < (x / 2)) -> (normh` ((F` z) -h (F` w))) < x))
47	6	ffvelrni 4788	. . . . . . . . . . . . 13 |- (w e. NN -> (F` w) e. ~H)
48	47, 26	jctil 316	. . . . . . . . . . . 12 |- (w e. NN -> (A e. ~H /\ (F` w) e. ~H))
49	46, 48	sylan2 500	. . . . . . . . . . 11 |- (((x e. RR /\ z e. NN) /\ w e. NN) -> (((normh` ((F` z) -h A)) < (x / 2) /\ (normh` ((F` w) -h A)) < (x / 2)) -> (normh` ((F` z) -h (F` w))) < x))
50	32, 49	syl9r 72	. . . . . . . . . 10 |- (((x e. RR /\ z e. NN) /\ w e. NN) -> (((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) -> ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
51	50	ralimdvaa 2171	. . . . . . . . 9 |- ((x e. RR /\ z e. NN) -> (A.w e. NN ((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) -> A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
52	51	ralimdvaa 2171	. . . . . . . 8 |- (x e. RR -> (A.z e. NN A.w e. NN ((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) -> A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
53		raaanv 2977	. . . . . . . . 9 |- (A.z e. NN A.w e. NN ((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) <-> (A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ A.w e. NN (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))))
54		breq2 3342	. . . . . . . . . . . 12 |- (w = z -> (y <_ w <-> y <_ z))
55		fveq2 4681	. . . . . . . . . . . . . . 15 |- (w = z -> (F` w) = (F` z))
56	55	opreq1d 4897	. . . . . . . . . . . . . 14 |- (w = z -> ((F` w) -h A) = ((F` z) -h A))
57	56	fveq2d 4685	. . . . . . . . . . . . 13 |- (w = z -> (normh` ((F` w) -h A)) = (normh` ((F` z) -h A)))
58	57	breq1d 3348	. . . . . . . . . . . 12 |- (w = z -> ((normh` ((F` w) -h A)) < (x / 2) <-> (normh` ((F` z) -h A)) < (x / 2)))
59	54, 58	imbi12d 688	. . . . . . . . . . 11 |- (w = z -> ((y <_ w -> (normh` ((F` w) -h A)) < (x / 2)) <-> (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
60	59	cbvralv 2280	. . . . . . . . . 10 |- (A.w e. NN (y <_ w -> (normh` ((F` w) -h A)) < (x / 2)) <-> A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)))
61	60	anbi2i 538	. . . . . . . . 9 |- ((A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ A.w e. NN (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) <-> (A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
62		anidm 478	. . . . . . . . 9 |- ((A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))) <-> A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)))
63	53, 61, 62	3bitri 194	. . . . . . . 8 |- (A.z e. NN A.w e. NN ((y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) /\ (y <_ w -> (normh` ((F` w) -h A)) < (x / 2))) <-> A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)))
64	52, 63	syl5ibr 224	. . . . . . 7 |- (x e. RR -> (A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) -> A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
65	64	reximdv 2202	. . . . . 6 |- (x e. RR -> (E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
66	65	adantr 425	. . . . 5 |- ((x e. RR /\ 0 < x) -> (E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < (x / 2)) -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
67	31, 66	mpd 29	. . . 4 |- ((x e. RR /\ 0 < x) -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x))
68	67	ex 402	. . 3 |- (x e. RR -> (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x)))
69	68	rgen 2159	. 2 |- A.x e. RR (0 < x -> E.y e. NN A.z e. NN A.w e. NN ((y <_ z /\ y <_ w) -> (normh` ((F` z) -h (F` w))) < x))
70	1, 6, 69	mpbir2an 800	1 |- F e. Cauchy

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292  ifcif 2982   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  RRcr 6385  0cc0 6386   / cdiv 6447   <_ cle 6448  NNcn 6449   < clt 6653  2c2 7145  ~Hchil 10420   -h cmv 10424  normhcno 10426  Cauchyccau 10427   ~~>v chli 10428

This theorem is referenced by:  hlimcaui 10740

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-hnorm 10469  df-hvsub 10472  df-hlim 10473  df-hcau 10474

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