Theorem hlimcauii 9382

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 9327	. 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 9333	. . 3 |- (F ~~>v A -> F:NN-->H~)
6	2, 5	ax-mp 7	. 2 |- F:NN-->H~
7		breq2 2696	. . . . . . . 8 |- (v = (x / 2) -> (0 < v <-> 0 < (x / 2)))
8		breq2 2696	. . . . . . . . . 10 |- (v = (x / 2) -> ((normh` ((F` z) -h A)) < v <-> (normh` ((F` z) -h A)) < (x / 2)))
9	8	imbi2d 615	. . . . . . . . 9 |- (v = (x / 2) -> ((y <_ z -> (normh` ((F` z) -h A)) < v) <-> (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
10	9	rexralbidv 1728	. . . . . . . 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))))
11	7, 10	imbi12d 629	. . . . . . 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)))))
12	3, 4	hlimveci 9334	. . . . . . . . 9 |- (F ~~>v A -> A e. H~)
13	2, 12	ax-mp 7	. . . . . . . 8 |- A e. H~
14		hlim2 9336	. . . . . . . . 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))))
15	2, 14	mpbii 191	. . . . . . . 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)))
16	6, 13, 15	mp2an 701	. . . . . . 7 |- A.v e. RR (0 < v -> E.y e. NN A.z e. NN (y <_ z -> (normh` ((F` z) -h A)) < v))
17	11, 16	vtoclri 1905	. . . . . 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))))
18		rehalfcl 6180	. . . . . . 7 |- (x e. RR -> (x / 2) e. RR)
19	18	adantr 389	. . . . . 6 |- ((x e. RR /\ 0 < x) -> (x / 2) e. RR)
20		breq2 2696	. . . . . . . . 9 |- (x = if(x e. RR, x, 0) -> (0 < x <-> 0 < if(x e. RR, x, 0)))
21		opreq1 4026	. . . . . . . . . 10 |- (x = if(x e. RR, x, 0) -> (x / 2) = (if(x e. RR, x, 0) / 2))
22	21	breq2d 2703	. . . . . . . . 9 |- (x = if(x e. RR, x, 0) -> (0 < (x / 2) <-> 0 < (if(x e. RR, x, 0) / 2)))
23	20, 22	imbi12d 629	. . . . . . . 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))))
24		0re 5594	. . . . . . . . . 10 |- 0 e. RR
25	24	elimel 2451	. . . . . . . . 9 |- if(x e. RR, x, 0) e. RR
26		2re 6125	. . . . . . . . 9 |- 2 e. RR
27		2pos 6135	. . . . . . . . 9 |- 0 < 2
28	25, 26, 27	divgt0i2i 6003	. . . . . . . 8 |- (0 < if(x e. RR, x, 0) -> 0 < (if(x e. RR, x, 0) / 2))
29	23, 28	dedth 2437	. . . . . . 7 |- (x e. RR -> (0 < x -> 0 < (x / 2)))
30	29	imp 348	. . . . . 6 |- ((x e. RR /\ 0 < x) -> 0 < (x / 2))
31	17, 19, 30	sylc 68	. . . . 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 559	. . . . . . . . . . 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 9286	. . . . . . . . . . . . . . . 16 |- ((A e. H~ /\ (F` w) e. H~) -> (normh` (A -h (F` w))) = (normh` ((F` w) -h A)))
34	33	breq1d 2702	. . . . . . . . . . . . . . 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 619	. . . . . . . . . . . . . 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 388	. . . . . . . . . . . . 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 3929	. . . . . . . . . . . . . . . . . . . 20 |- (z e. NN -> (F` z) e. H~)
38	37	anim1i 332	. . . . . . . . . . . . . . . . . . 19 |- ((z e. NN /\ (F` w) e. H~) -> ((F` z) e. H~ /\ (F` w) e. H~))
39	38	ancoms 438	. . . . . . . . . . . . . . . . . 18 |- (((F` w) e. H~ /\ z e. NN) -> ((F` z) e. H~ /\ (F` w) e. H~))
40	39	anim1i 332	. . . . . . . . . . . . . . . . 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 438	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . 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 438	. . . . . . . . . . . . . 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 9295	. . . . . . . . . . . . . 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 10	. . . . . . . . . . . . 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 203	. . . . . . . . . . . 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 3929	. . . . . . . . . . . . 13 |- (w e. NN -> (F` w) e. H~)
48	47, 13	jctil 290	. . . . . . . . . . . 12 |- (w e. NN -> (A e. H~ /\ (F` w) e. H~))
49	46, 48	sylan2 453	. . . . . . . . . . 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 58	. . . . . . . . . 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	r19.20dva 1755	. . . . . . . . 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	r19.20dva 1755	. . . . . . . 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		raaan 2414	. . . . . . . . 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 2696	. . . . . . . . . . . 12 |- (w = z -> (y <_ w <-> y <_ z))
55		fveq2 3835	. . . . . . . . . . . . . . 15 |- (w = z -> (F` w) = (F` z))
56	55	opreq1d 4033	. . . . . . . . . . . . . 14 |- (w = z -> ((F` w) -h A) = ((F` z) -h A))
57	56	fveq2d 3839	. . . . . . . . . . . . 13 |- (w = z -> (normh` ((F` w) -h A)) = (normh` ((F` z) -h A)))
58	57	breq1d 2702	. . . . . . . . . . . 12 |- (w = z -> ((normh` ((F` w) -h A)) < (x / 2) <-> (normh` ((F` z) -h A)) < (x / 2)))
59	54, 58	imbi12d 629	. . . . . . . . . . 11 |- (w = z -> ((y <_ w -> (normh` ((F` w) -h A)) < (x / 2)) <-> (y <_ z -> (normh` ((F` z) -h A)) < (x / 2))))
60	59	cbvralv 1846	. . . . . . . . . 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 483	. . . . . . . . 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 433	. . . . . . . . 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 175	. . . . . . . 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 205	. . . . . . 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	r19.22sdv 1784	. . . . . 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 389	. . . . 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 26	. . . 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 371	. . 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 1744	. 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 735	1 |- F e. Cauchy

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  E.wrex 1692  Vcvv 1857  ifcif 2415   class class class wbr 2692  -->wf 3259  ` cfv 3263  (class class class)co 4021  RRcr 5387  0cc0 5388   / cdiv 5448   <_ cle 5449  NNcn 5450   < clt 5640  2c2 6107  H~chil 9063   -h cmv 9067  normhcno 9069  Cauchyccau 9070   ~~>v chli 9071

This theorem is referenced by:  hlimcaui 9383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-hfvadd 9145  ax-hvcom 9146  ax-hvass 9147  ax-hv0cl 9148  ax-hvaddid 9149  ax-hfvmul 9150  ax-hvmulid 9151  ax-hvmulass 9152  ax-hvdistr1 9153  ax-hvdistr2 9154  ax-hvmul0 9155  ax-hfi 9222  ax-his1 9225  ax-his2 9226  ax-his3 9227  ax-his4 9228

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-sup 4717  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-2 6116  df-3 6117  df-4 6118  df-n0 6268  df-z 6304  df-seq1 6673  df-exp 6764  df-sqr 6871  df-re 6952  df-im 6953  df-cj 6954  df-abs 6955  df-hnorm 9112  df-hvsub 9115  df-hlim 9116  df-hcau 9117

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