Theorem chcmh 9034

Description: The hypothesis defines the set of complete subspaces of Hilbert space (see chsscm 9033). 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 groups:   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 9033	. 2 |- CH (_ C
3		df-ral 1641	. . . . . 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-17 968	. . . . . . . . 9 |- (f e. Cauchy -> A.x f e. Cauchy)
5		ax-17 968	. . . . . . . . . 10 |- (f:NN-->h -> A.x f:NN-->h)
6		hbre1 1681	. . . . . . . . . 10 |- (E.x e. h f ~~>v x -> A.xE.x e. h f ~~>v x)
7	5, 6	hbim 1004	. . . . . . . . 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 1004	. . . . . . . 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 1804	. . . . . . . . . . . . 13 |- x e. V
10		visset 1804	. . . . . . . . . . . . 13 |- f e. V
11	9, 10	hlimcau 9028	. . . . . . . . . . . 12 |- (f ~~>v x -> f e. Cauchy)
12	11	imim1i 16	. . . . . . . . . . 11 |- ((f e. Cauchy -> E.x e. h f ~~>v x) -> (f ~~>v x -> E.x e. h f ~~>v x))
13		19.8a 1025	. . . . . . . . . . . . . 14 |- (f ~~>v x -> E.x f ~~>v x)
14	10	hlimeu 9032	. . . . . . . . . . . . . 14 |- (E.x f ~~>v x <-> E!x f ~~>v x)
15	13, 14	sylib 198	. . . . . . . . . . . . 13 |- (f ~~>v x -> E!x f ~~>v x)
16		eupick 1427	. . . . . . . . . . . . . . 15 |- ((E!x f ~~>v x /\ E.x(f ~~>v x /\ x e. h)) -> (f ~~>v x -> x e. h))
17	16	ex 373	. . . . . . . . . . . . . 14 |- (E!x f ~~>v x -> (E.x(f ~~>v x /\ x e. h) -> (f ~~>v x -> x e. h)))
18		df-rex 1642	. . . . . . . . . . . . . . 15 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
19		exancom 1050	. . . . . . . . . . . . . . 15 |- (E.x(x e. h /\ f ~~>v x) <-> E.x(f ~~>v x /\ x e. h))
20	18, 19	bitr 173	. . . . . . . . . . . . . 14 |- (E.x e. h f ~~>v x <-> E.x(f ~~>v x /\ x e. h))
21	17, 20	syl5ib 206	. . . . . . . . . . . . 13 |- (E!x f ~~>v x -> (E.x e. h f ~~>v x -> (f ~~>v x -> x e. h)))
22	15, 21	syl 10	. . . . . . . . . . . 12 |- (f ~~>v x -> (E.x e. h f ~~>v x -> (f ~~>v x -> x e. h)))
23	22	pm2.43a 66	. . . . . . . . . . 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	imim2i 17	. . . . . . . . 9 |- ((f:NN-->h -> (f e. Cauchy -> E.x e. h f ~~>v x)) -> (f:NN-->h -> (f ~~>v x -> x e. h)))
26		bi2.04 160	. . . . . . . . 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 347	. . . . . . . . 9 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
28	25, 26, 27	3imtr4 219	. . . . . . . 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 995	. . . . . . 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 989	. . . . . 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 199	. . . . 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 335	. . . 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	abeq2i 1562	. . . 4 |- (h e. C <-> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
34		closedsub 9014	. . . 4 |- (h e. CH <-> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
35	32, 33, 34	3imtr4 219	. . 3 |- (h e. C -> h e. CH)
36	35	ssriv 2059	. 2 |- C (_ CH
37	2, 36	eqssi 2068	1 |- CH = C

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  {cab 1456  A.wral 1637  E.wrex 1638   class class class wbr 2609  -->wf 3168  NNcn 5268  Cauchyccau 8734   ~~>v chli 8735  SHcsh 8736  CHcch 8737

This theorem is referenced by:  ch2 9035

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-inf2 4597  ax-hfvadd 8791  ax-hvcom 8792  ax-hvass 8793  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvmulass 8798  ax-hvdistr1 8799  ax-hvdistr2 8800  ax-hvmul0 8801  ax-hfi 8867  ax-his1 8870  ax-his2 8871  ax-his3 8872  ax-his4 8873  ax-hcompl 8992

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-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-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-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  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-seq1 6245  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-hnorm 8776  df-hvsub 8779  df-hlim 8780  df-hcau 8781  df-ch 9013

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