Theorem chcmh 5148

Description: The hypothesis defines the set of complete subspaces of Hilbert space (see chsscm 5147). A Hilbert subspace is closed iff it is complete. Remark 3.12(C) of [Beran] p. 107.

Hypothesis

Ref	Expression
cmh.1	|- C = {h | (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))}

Assertion

Ref	Expression
chcmh	|- CH = C

Distinct variable group(s):   x,f,h   C,h

Proof of Theorem chcmh

Step	Hyp	Ref	Expression
1		cmh.1	. . 3 |- C = {h | (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x))}
2	1	chsscm 5147	. 2 |- CH (_ C
3		df-ral 1205	. . . . . 6 |- (A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x) <-> A.f(f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)))
4		ax-17 925	. . . . . . . . 9 |- (f e. Cauchy -> A.x f e. Cauchy)
5		ax-17 925	. . . . . . . . . 10 |- (f:NN-->h -> A.x f:NN-->h)
6		hbre1 1239	. . . . . . . . . 10 |- (E.x e. h f ~~>v x -> A.xE.x e. h f ~~>v x)
7	5, 6	hbim 702	. . . . . . . . 9 |- ((f:NN-->h -> E.x e. h f ~~>v x) -> A.x(f:NN-->h -> E.x e. h f ~~>v x))
8	4, 7	hbim 702	. . . . . . . 8 |- ((f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)) -> A.x(f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)))
9		visset 1350	. . . . . . . . . . . . 13 |- x e. V
10		visset 1350	. . . . . . . . . . . . 13 |- f e. V
11	9, 10	hlimcau 5142	. . . . . . . . . . . 12 |- (f ~~>v x -> f e. Cauchy)
12	11	syl4 19	. . . . . . . . . . 11 |- ((f e. Cauchy -> E.x e. h f ~~>v x) -> (f ~~>v x -> E.x e. h f ~~>v x))
13		19.8a 712	. . . . . . . . . . . . . 14 |- (f ~~>v x -> E.x f ~~>v x)
14	10	hlimeu 5146	. . . . . . . . . . . . . 14 |- (E.x f ~~>v x <-> E!x f ~~>v x)
15	13, 14	sylib 173	. . . . . . . . . . . . 13 |- (f ~~>v x -> E!x f ~~>v x)
16		eupick 1055	. . . . . . . . . . . . . . 15 |- ((E!x f ~~>v x /\ E.x(f ~~>v x /\ x e. h)) -> (f ~~>v x -> x e. h))
17	16	exp 291	. . . . . . . . . . . . . 14 |- (E!x f ~~>v x -> (E.x(f ~~>v x /\ x e. h) -> (f ~~>v x -> x e. h)))
18		df-rex 1206	. . . . . . . . . . . . . . 15 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
19		exancom 736	. . . . . . . . . . . . . . 15 |- (E.x(x e. h /\ f ~~>v x) <-> E.x(f ~~>v x /\ x e. h))
20	18, 19	bitr 151	. . . . . . . . . . . . . 14 |- (E.x e. h f ~~>v x <-> E.x(f ~~>v x /\ x e. h))
21	17, 20	syl5ib 181	. . . . . . . . . . . . 13 |- (E!x f ~~>v x -> (E.x e. h f ~~>v x -> (f ~~>v x -> x e. h)))
22	15, 21	syl 12	. . . . . . . . . . . 12 |- (f ~~>v x -> (E.x e. h f ~~>v x -> (f ~~>v x -> x e. h)))
23	22	pm2.43a 60	. . . . . . . . . . 11 |- (f ~~>v x -> (E.x e. h f ~~>v x -> x e. h))
24	12, 23	sylcom 51	. . . . . . . . . 10 |- ((f e. Cauchy -> E.x e. h f ~~>v x) -> (f ~~>v x -> x e. h))
25	24	syl3 18	. . . . . . . . 9 |- ((f:NN-->h -> (f e. Cauchy -> E.x e. h f ~~>v x)) -> (f:NN-->h -> (f ~~>v x -> x e. h)))
26		bi2.04 141	. . . . . . . . 9 |- ((f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)) <-> (f:NN-->h -> (f e. Cauchy -> E.x e. h f ~~>v x)))
27		impexp 276	. . . . . . . . 9 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
28	25, 26, 27	3imtr4 192	. . . . . . . 8 |- ((f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)) -> ((f:NN-->h /\ f ~~>v x) -> x e. h))
29	8, 28	19.21ai 740	. . . . . . 7 |- ((f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)) -> A.x((f:NN-->h /\ f ~~>v x) -> x e. h))
30	29	19.20i 691	. . . . . 6 |- (A.f(f e. Cauchy -> (f:NN-->h -> E.x e. h f ~~>v x)) -> A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))
31	3, 30	sylbi 174	. . . . 5 |- (A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x) -> A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h))
32	31	anim2i 270	. . . 4 |- ((h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)) -> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
33	1	cleqabi 1176	. . . 4 |- (h e. C <-> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
34		closedsub 5128	. . . 4 |- (h e. CH <-> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
35	32, 33, 34	3imtr4 192	. . 3 |- (h e. C -> h e. CH)
36	35	ssriv 1508	. 2 |- C (_ CH
37	2, 36	eqssi 1517	1 |- CH = C

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   class class class wbr 2054  -->wf 2418  NNcn 4093  Cauchyccau 4965   ~~>v chli 4966  SHcsh 4967  CHcch 4968

This theorem is referenced by:  ch2 5149

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107  df-ch 5127

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