Theorem chcmhi 10746

Description: The hypothesis defines the set of complete subspaces of Hilbert space (see chsscmi 10745). 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
chcmhi	|- CH = C

Distinct variable groups:   x,f,h   C,h

Proof of Theorem chcmhi

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	chsscmi 10745	. 2 |- CH C_ C
3		df-ral 2109	. . . . . 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 1317	. . . . . . . . 9 |- (f e. Cauchy -> A.x f e. Cauchy)
5		ax-17 1317	. . . . . . . . . 10 |- (f:NN-->h -> A.x f:NN-->h)
6		hbre1 2150	. . . . . . . . . 10 |- (E.x e. h f ~~>v x -> A.xE.x e. h f ~~>v x)
7	5, 6	hbim 1354	. . . . . . . . 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 1354	. . . . . . . 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 2295	. . . . . . . . . . . . 13 |- x e. _V
10		visset 2295	. . . . . . . . . . . . 13 |- f e. _V
11	9, 10	hlimcaui 10740	. . . . . . . . . . . 12 |- (f ~~>v x -> f e. Cauchy)
12	11	imim1i 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 1376	. . . . . . . . . . . . . 14 |- (f ~~>v x -> E.x f ~~>v x)
14	10	hlimeui 10744	. . . . . . . . . . . . . 14 |- (E.x f ~~>v x <-> E!x f ~~>v x)
15	13, 14	sylib 215	. . . . . . . . . . . . 13 |- (f ~~>v x -> E!x f ~~>v x)
16		eupick 1834	. . . . . . . . . . . . . . 15 |- ((E!x f ~~>v x /\ E.x(f ~~>v x /\ x e. h)) -> (f ~~>v x -> x e. h))
17	16	ex 402	. . . . . . . . . . . . . 14 |- (E!x f ~~>v x -> (E.x(f ~~>v x /\ x e. h) -> (f ~~>v x -> x e. h)))
18		df-rex 2110	. . . . . . . . . . . . . . 15 |- (E.x e. h f ~~>v x <-> E.x(x e. h /\ f ~~>v x))
19		exancom 1401	. . . . . . . . . . . . . . 15 |- (E.x(x e. h /\ f ~~>v x) <-> E.x(f ~~>v x /\ x e. h))
20	18, 19	bitri 190	. . . . . . . . . . . . . 14 |- (E.x e. h f ~~>v x <-> E.x(f ~~>v x /\ x e. h))
21	17, 20	syl5ib 223	. . . . . . . . . . . . 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 80	. . . . . . . . . . 11 |- (f ~~>v x -> (E.x e. h f ~~>v x -> x e. h))
24	12, 23	sylcom 62	. . . . . . . . . 10 |- ((f e. Cauchy -> E.x e. h f ~~>v x) -> (f ~~>v x -> x e. h))
25	24	imim2i 11	. . . . . . . . 9 |- ((f:NN-->h -> (f e. Cauchy -> E.x e. h f ~~>v x)) -> (f:NN-->h -> (f ~~>v x -> x e. h)))
26		bi2.04 177	. . . . . . . . 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 374	. . . . . . . . 9 |- (((f:NN-->h /\ f ~~>v x) -> x e. h) <-> (f:NN-->h -> (f ~~>v x -> x e. h)))
28	25, 26, 27	3imtr4i 236	. . . . . . . 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 1345	. . . . . . 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	alimi 1338	. . . . . 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 216	. . . . 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 362	. . . 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 2001	. . . 4 |- (h e. C <-> (h e. SH /\ A.f e. Cauchy (f:NN-->h -> E.x e. h f ~~>v x)))
34		closedsub 10726	. . . 4 |- (h e. CH <-> (h e. SH /\ A.fA.x((f:NN-->h /\ f ~~>v x) -> x e. h)))
35	32, 33, 34	3imtr4i 236	. . 3 |- (h e. C -> h e. CH)
36	35	ssriv 2621	. 2 |- C C_ CH
37	2, 36	eqssi 2632	1 |- CH = C

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  A.wral 2105  E.wrex 2106   class class class wbr 3338  -->wf 3994  NNcn 6449  Cauchyccau 10427   ~~>v chli 10428  SHcsh 10429  CHcch 10430

This theorem is referenced by:  ch2 10747

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  ax-hcompl 10704

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  df-ch 10725

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