Theorem climi 11433

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 4034	. . . 4 |- ( x = C -> ( ( abs ` ( B - A ) ) < x <-> ( abs ` ( B - A ) ) < C ) )
2	1	anbi2d 464	. . 3 |- ( x = C -> ( ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < C ) ) )
3	2	rexralbidv 2520	. 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 11426	. . . . . . 7 |- Rel ~~>
8	7	brrelex1i 4703	. . . . . 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 11429	. . . 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 147	. . 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 114	. 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 2870	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   < clt 8056   - cmin 8192   ZZcz 9320   ZZ>=cuz 9595   RR+crp 9722   abscabs 11144   ~~> cli 11424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-clim 11425

This theorem is referenced by:  climi2  11434  climi0  11435  climuni  11439  2clim  11447  climcau  11493  climcaucn  11497