Theorem climi 11540

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 4047	. . . 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 2531	. 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 11533	. . . . . . 7 |- Rel ~~>
8	7	brrelex1i 4717	. . . . . 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 11536	. . . 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 2881	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 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   _Vcvv 2771   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   CCcc 7922   < clt 8106   - cmin 8242   ZZcz 9371   ZZ>=cuz 9647   RR+crp 9774   abscabs 11250   ~~> cli 11531

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-clim 11532

This theorem is referenced by:  climi2  11541  climi0  11542  climuni  11546  2clim  11554  climcau  11600  climcaucn  11604