Theorem chlimi 10737

Description: The limit property of a closed subspace of a Hilbert space.

Hypothesis

Ref	Expression
chlim.1	|- A e. _V

Assertion

Ref	Expression
chlimi	|- ((H e. CH /\ F:NN-->H /\ F ~~>v A) -> A e. H)

Proof of Theorem chlimi

Step	Hyp	Ref	Expression
1		closedsub 10726	. . . 4 |- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))
2	1	simprbi 353	. . 3 |- (H e. CH -> A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H))
3		nnex 7116	. . . . . . 7 |- NN e. _V
4		fex 4595	. . . . . . 7 |- ((F:NN-->H /\ NN e. _V) -> F e. _V)
5	3, 4	mpan2 760	. . . . . 6 |- (F:NN-->H -> F e. _V)
6	5	adantr 425	. . . . 5 |- ((F:NN-->H /\ F ~~>v A) -> F e. _V)
7		feq1 4551	. . . . . . . . . 10 |- (f = F -> (f:NN-->H <-> F:NN-->H))
8		breq1 3341	. . . . . . . . . 10 |- (f = F -> (f ~~>v x <-> F ~~>v x))
9	7, 8	anbi12d 690	. . . . . . . . 9 |- (f = F -> ((f:NN-->H /\ f ~~>v x) <-> (F:NN-->H /\ F ~~>v x)))
10	9	imbi1d 675	. . . . . . . 8 |- (f = F -> (((f:NN-->H /\ f ~~>v x) -> x e. H) <-> ((F:NN-->H /\ F ~~>v x) -> x e. H)))
11	10	albidv 1656	. . . . . . 7 |- (f = F -> (A.x((f:NN-->H /\ f ~~>v x) -> x e. H) <-> A.x((F:NN-->H /\ F ~~>v x) -> x e. H)))
12	11	cla4gv 2364	. . . . . 6 |- (F e. _V -> (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> A.x((F:NN-->H /\ F ~~>v x) -> x e. H)))
13		chlim.1	. . . . . . 7 |- A e. _V
14		breq2 3342	. . . . . . . . 9 |- (x = A -> (F ~~>v x <-> F ~~>v A))
15	14	anbi2d 678	. . . . . . . 8 |- (x = A -> ((F:NN-->H /\ F ~~>v x) <-> (F:NN-->H /\ F ~~>v A)))
16		eleq1 1957	. . . . . . . 8 |- (x = A -> (x e. H <-> A e. H))
17	15, 16	imbi12d 688	. . . . . . 7 |- (x = A -> (((F:NN-->H /\ F ~~>v x) -> x e. H) <-> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
18	13, 17	cla4v 2370	. . . . . 6 |- (A.x((F:NN-->H /\ F ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
19	12, 18	syl6 25	. . . . 5 |- (F e. _V -> (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
20	6, 19	syl 12	. . . 4 |- ((F:NN-->H /\ F ~~>v A) -> (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H)))
21	20	pm2.43b 81	. . 3 |- (A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H) -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
22	2, 21	syl 12	. 2 |- (H e. CH -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
23	22	3impib 1065	1 |- ((H e. CH /\ F:NN-->H /\ F ~~>v A) -> A e. H)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  -->wf 3994  NNcn 6449   ~~>v chli 10428  SHcsh 10429  CHcch 10430

This theorem is referenced by:  chintcli 10928  osumlem6 11218

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

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-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-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-sub 6511  df-neg 6513  df-n 7108  df-ch 10725

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