Theorem lmle 23099

Description: If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
lmle.1	|- Z = ( ZZ>= ` M )
lmle.3	|- J = ( MetOpen ` D )
lmle.4	|- ( ph -> D e. ( *Met ` X ) )
lmle.6	|- ( ph -> M e. ZZ )
lmle.7	|- ( ph -> F ( ~~> t ` J ) P )
lmle.8	|- ( ph -> Q e. X )
lmle.9	|- ( ph -> R e. RR* )
lmle.10	|- ( ( ph /\ k e. Z ) -> ( Q D ( F ` k ) ) <_ R )

Assertion

Ref	Expression
lmle	|- ( ph -> ( Q D P ) <_ R )

Distinct variable groups:   D, k   k, J   ph, k   k, Z   k, F   P, k   Q, k   R, k   k, X

Allowed substitution hint:   M( k)

Proof of Theorem lmle

Dummy variables j x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmle.1	. . . 4 |- Z = ( ZZ>= ` M )
2		lmle.4	. . . . 5 |- ( ph -> D e. ( *Met ` X ) )
3		lmle.3	. . . . . 6 |- J = ( MetOpen ` D )
4	3	mopntopon 22244	. . . . 5 |- ( D e. ( *Met ` X ) -> J e. (TopOn ` X ) )
5	2, 4	syl 17	. . . 4 |- ( ph -> J e. (TopOn ` X ) )
6		lmle.6	. . . 4 |- ( ph -> M e. ZZ )
7		lmrel 21034	. . . . 5 |- Rel ( ~~> t ` J )
8		lmle.7	. . . . 5 |- ( ph -> F ( ~~> t ` J ) P )
9		releldm 5358	. . . . 5 |- ( ( Rel ( ~~> t ` J ) /\ F ( ~~> t ` J ) P ) -> F e. dom ( ~~> t ` J ) )
10	7, 8, 9	sylancr 695	. . . 4 |- ( ph -> F e. dom ( ~~> t ` J ) )
11	1, 5, 6, 10	lmff 21105	. . 3 |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
12		eqid 2622	. . . 4 |- ( ZZ>= ` j ) = ( ZZ>= ` j )
13	5	adantr 481	. . . 4 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> J e. (TopOn ` X ) )
14		simprl 794	. . . . . 6 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> j e. Z )
15	14, 1	syl6eleq 2711	. . . . 5 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> j e. ( ZZ>= ` M ) )
16		eluzelz 11697	. . . . 5 |- ( j e. ( ZZ>= ` M ) -> j e. ZZ )
17	15, 16	syl 17	. . . 4 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> j e. ZZ )
18	8	adantr 481	. . . 4 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> F ( ~~> t ` J ) P )
19		fvres 6207	. . . . . . 7 |- ( k e. ( ZZ>= ` j ) -> ( ( F |` ( ZZ>= ` j ) ) ` k ) = ( F ` k ) )
20	19	adantl 482	. . . . . 6 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( ( F |` ( ZZ>= ` j ) ) ` k ) = ( F ` k ) )
21		simprr 796	. . . . . . 7 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
22	21	ffvelrnda 6359	. . . . . 6 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( ( F |` ( ZZ>= ` j ) ) ` k ) e. X )
23	20, 22	eqeltrrd 2702	. . . . 5 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( F ` k ) e. X )
24	1	uztrn2 11705	. . . . . . 7 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
25	14, 24	sylan 488	. . . . . 6 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
26		lmle.10	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( Q D ( F ` k ) ) <_ R )
27	26	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. Z ) -> ( Q D ( F ` k ) ) <_ R )
28	25, 27	syldan 487	. . . . 5 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( Q D ( F ` k ) ) <_ R )
29		oveq2 6658	. . . . . . 7 |- ( x = ( F ` k ) -> ( Q D x ) = ( Q D ( F ` k ) ) )
30	29	breq1d 4663	. . . . . 6 |- ( x = ( F ` k ) -> ( ( Q D x ) <_ R <-> ( Q D ( F ` k ) ) <_ R ) )
31	30	elrab 3363	. . . . 5 |- ( ( F ` k ) e. { x e. X | ( Q D x ) <_ R } <-> ( ( F ` k ) e. X /\ ( Q D ( F ` k ) ) <_ R ) )
32	23, 28, 31	sylanbrc 698	. . . 4 |- ( ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) /\ k e. ( ZZ>= ` j ) ) -> ( F ` k ) e. { x e. X | ( Q D x ) <_ R } )
33		lmle.8	. . . . . 6 |- ( ph -> Q e. X )
34		lmle.9	. . . . . 6 |- ( ph -> R e. RR* )
35		eqid 2622	. . . . . . 7 |- { x e. X | ( Q D x ) <_ R } = { x e. X | ( Q D x ) <_ R }
36	3, 35	blcld 22310	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ Q e. X /\ R e. RR* ) -> { x e. X | ( Q D x ) <_ R } e. ( Clsd ` J ) )
37	2, 33, 34, 36	syl3anc 1326	. . . . 5 |- ( ph -> { x e. X | ( Q D x ) <_ R } e. ( Clsd ` J ) )
38	37	adantr 481	. . . 4 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> { x e. X | ( Q D x ) <_ R } e. ( Clsd ` J ) )
39	12, 13, 17, 18, 32, 38	lmcld 21107	. . 3 |- ( ( ph /\ ( j e. Z /\ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) ) -> P e. { x e. X | ( Q D x ) <_ R } )
40	11, 39	rexlimddv 3035	. 2 |- ( ph -> P e. { x e. X | ( Q D x ) <_ R } )
41		oveq2 6658	. . . . 5 |- ( x = P -> ( Q D x ) = ( Q D P ) )
42	41	breq1d 4663	. . . 4 |- ( x = P -> ( ( Q D x ) <_ R <-> ( Q D P ) <_ R ) )
43	42	elrab 3363	. . 3 |- ( P e. { x e. X | ( Q D x ) <_ R } <-> ( P e. X /\ ( Q D P ) <_ R ) )
44	43	simprbi 480	. 2 |- ( P e. { x e. X | ( Q D x ) <_ R } -> ( Q D P ) <_ R )
45	40, 44	syl 17	1 |- ( ph -> ( Q D P ) <_ R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   dom cdm 5114   |` cres 5116   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   RR*cxr 10073   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687   *Metcxmt 19731   MetOpencmopn 19736  TopOnctopon 20715   Clsdccld 20820   ~~> tclm 21030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033

This theorem is referenced by:  nglmle  23100  minvecolem4  27736