Theorem climi 11309

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 4019	. . . 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 2513	. 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 11302	. . . . . . 7 |- Rel ~~>
8	7	brrelex1i 4681	. . . . . 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 11305	. . . 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 2858	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 1363   e. wcel 2158   A.wral 2465   E.wrex 2466   _Vcvv 2749   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   CCcc 7823   < clt 8006   - cmin 8142   ZZcz 9267   ZZ>=cuz 9542   RR+crp 9667   abscabs 11020   ~~> cli 11300

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-clim 11301

This theorem is referenced by:  climi2  11310  climi0  11311  climuni  11315  2clim  11323  climcau  11369  climcaucn  11373