Theorem chlim 9104

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

Hypothesis

Ref	Expression
chlim.1	|- A e. V

Assertion

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

Proof of Theorem chlim

Step	Hyp	Ref	Expression
1		closedsub 9093	. . . 4 |- (H e. CH <-> (H e. SH /\ A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H)))
2	1	pm3.27bi 326	. . 3 |- (H e. CH -> A.fA.x((f:NN-->H /\ f ~~>v x) -> x e. H))
3		nnex 5933	. . . . . . 7 |- NN e. V
4		fex 3652	. . . . . . 7 |- ((F:NN-->H /\ NN e. V) -> F e. V)
5	3, 4	mpan2 696	. . . . . 6 |- (F:NN-->H -> F e. V)
6	5	adantr 389	. . . . 5 |- ((F:NN-->H /\ F ~~>v A) -> F e. V)
7		feq1 3620	. . . . . . . . . 10 |- (f = F -> (f:NN-->H <-> F:NN-->H))
8		breq1 2622	. . . . . . . . . 10 |- (f = F -> (f ~~>v x <-> F ~~>v x))
9	7, 8	anbi12d 628	. . . . . . . . 9 |- (f = F -> ((f:NN-->H /\ f ~~>v x) <-> (F:NN-->H /\ F ~~>v x)))
10	9	imbi1d 613	. . . . . . . 8 |- (f = F -> (((f:NN-->H /\ f ~~>v x) -> x e. H) <-> ((F:NN-->H /\ F ~~>v x) -> x e. H)))
11	10	albidv 1278	. . . . . . 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 1862	. . . . . 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 2623	. . . . . . . . 9 |- (x = A -> (F ~~>v x <-> F ~~>v A))
15	14	anbi2d 616	. . . . . . . 8 |- (x = A -> ((F:NN-->H /\ F ~~>v x) <-> (F:NN-->H /\ F ~~>v A)))
16		eleq1 1534	. . . . . . . 8 |- (x = A -> (x e. H <-> A e. H))
17	15, 16	imbi12d 626	. . . . . . 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 1868	. . . . . 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 22	. . . . 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 10	. . . 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 67	. . 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 10	. 2 |- (H e. CH -> ((F:NN-->H /\ F ~~>v A) -> A e. H))
23	22	3impib 831	1 |- ((H e. CH /\ F:NN-->H /\ F ~~>v A) -> A e. H)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  -->wf 3178  NNcn 5296   ~~>v chli 8796  SHcsh 8797  CHcch 8798

This theorem is referenced by:  chintcl 9295  osumlem6 9583

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-n 5925  df-ch 9092

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