Theorem lmmcvg 23059

Description: Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
lmmbr.2	|- J = ( MetOpen ` D )
lmmbr.3	|- ( ph -> D e. ( *Met ` X ) )
lmmbr3.5	|- Z = ( ZZ>= ` M )
lmmbr3.6	|- ( ph -> M e. ZZ )
lmmbrf.7	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
lmmcvg.8	|- ( ph -> F ( ~~> t ` J ) P )
lmmcvg.9	|- ( ph -> R e. RR+ )

Assertion

Ref	Expression
lmmcvg	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( A e. X /\ ( A D P ) < R ) )

Distinct variable groups:   j, k, D   j, F, k   P, j, k   j, X, k   j, M   ph, j, k   R, j, k   j, Z, k

Allowed substitution hints:   A( j, k)   J( j, k)   M( k)

Proof of Theorem lmmcvg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmmcvg.9	. . 3 |- ( ph -> R e. RR+ )
2		lmmcvg.8	. . . . 5 |- ( ph -> F ( ~~> t ` J ) P )
3		lmmbr.2	. . . . . 6 |- J = ( MetOpen ` D )
4		lmmbr.3	. . . . . 6 |- ( ph -> D e. ( *Met ` X ) )
5		lmmbr3.5	. . . . . 6 |- Z = ( ZZ>= ` M )
6		lmmbr3.6	. . . . . 6 |- ( ph -> M e. ZZ )
7	3, 4, 5, 6	lmmbr3 23058	. . . . 5 |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) ) ) )
8	2, 7	mpbid 222	. . . 4 |- ( ph -> ( F e. ( X ^pm CC ) /\ P e. X /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) ) )
9	8	simp3d 1075	. . 3 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) )
10		breq2 4657	. . . . . 6 |- ( x = R -> ( ( ( F ` k ) D P ) < x <-> ( ( F ` k ) D P ) < R ) )
11	10	3anbi3d 1405	. . . . 5 |- ( x = R -> ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) <-> ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) ) )
12	11	rexralbidv 3058	. . . 4 |- ( x = R -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) ) )
13	12	rspcv 3305	. . 3 |- ( R e. RR+ -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < x ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) ) )
14	1, 9, 13	sylc 65	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) )
15	5	uztrn2 11705	. . . . . 6 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
16		3simpc 1060	. . . . . . 7 |- ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> ( ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) )
17		lmmbrf.7	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
18	17	eleq1d 2686	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) e. X <-> A e. X ) )
19	17	oveq1d 6665	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) D P ) = ( A D P ) )
20	19	breq1d 4663	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( ( ( F ` k ) D P ) < R <-> ( A D P ) < R ) )
21	18, 20	anbi12d 747	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) <-> ( A e. X /\ ( A D P ) < R ) ) )
22	16, 21	syl5ib 234	. . . . . 6 |- ( ( ph /\ k e. Z ) -> ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> ( A e. X /\ ( A D P ) < R ) ) )
23	15, 22	sylan2 491	. . . . 5 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> ( A e. X /\ ( A D P ) < R ) ) )
24	23	anassrs 680	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> ( A e. X /\ ( A D P ) < R ) ) )
25	24	ralimdva 2962	. . 3 |- ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> A. k e. ( ZZ>= ` j ) ( A e. X /\ ( A D P ) < R ) ) )
26	25	reximdva 3017	. 2 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. X /\ ( ( F ` k ) D P ) < R ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( A e. X /\ ( A D P ) < R ) ) )
27	14, 26	mpd 15	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( A e. X /\ ( A D P ) < R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   *Metcxmt 19731   MetOpencmopn 19736   ~~> 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-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-iun 4522  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-lm 21033

This theorem is referenced by:  bfplem2  33622