Theorem climi 11088

Description: Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
climi.1	|- Z = ( ZZ>= ` M )
climi.2	|- ( ph -> M e. ZZ )
climi.3	|- ( ph -> C e. RR+ )
climi.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
climi.5	|- ( ph -> F ~~> A )

Assertion

Ref	Expression
climi	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )

Distinct variable groups:   j, k, A   C, j, k   j, F, k   ph, j, k   j, Z, k   j, M

Allowed substitution hints:   B( j, k)   M( k)

Proof of Theorem climi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3941	. . . 4 |- ( x = C -> ( ( abs ` ( B - A ) ) < x <-> ( abs ` ( B - A ) ) < C ) )
2	1	anbi2d 460	. . 3 |- ( x = C -> ( ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < C ) ) )
3	2	rexralbidv 2464	. 2 |- ( x = C -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) ) )
4		climi.5	. . . 4 |- ( ph -> F ~~> A )
5		climi.1	. . . . 5 |- Z = ( ZZ>= ` M )
6		climi.2	. . . . 5 |- ( ph -> M e. ZZ )
7		climrel 11081	. . . . . . 7 |- Rel ~~>
8	7	brrelex1i 4590	. . . . . 6 |- ( F ~~> A -> F e. _V )
9	4, 8	syl 14	. . . . 5 |- ( ph -> F e. _V )
10		climi.4	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
11	5, 6, 9, 10	clim2 11084	. . . 4 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
12	4, 11	mpbid 146	. . 3 |- ( ph -> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
13	12	simprd 113	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) )
14		climi.3	. 2 |- ( ph -> C e. RR+ )
15	3, 13, 14	rspcdva 2798	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   A.wral 2417   E.wrex 2418   _Vcvv 2689   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   < clt 7824   - cmin 7957   ZZcz 9078   ZZ>=cuz 9350   RR+crp 9470   abscabs 10801   ~~> cli 11079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-clim 11080

This theorem is referenced by:  climi2  11089  climi0  11090  climuni  11094  2clim  11102  climcau  11148  climcaucn  11152