Theorem climi0 11999

Description: Convergence of a sequence of complex numbers to zero. (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 )
climi0.5	|- ( ph -> F ~~> 0 )

Assertion

Ref	Expression
climi0	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )

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

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

Proof of Theorem climi0

Step	Hyp	Ref	Expression
1		climi.1	. . 3 |- Z = ( ZZ>= ` M )
2		climi.2	. . 3 |- ( ph -> M e. ZZ )
3		climi.3	. . 3 |- ( ph -> C e. RR+ )
4		climi.4	. . 3 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
5		climi0.5	. . 3 |- ( ph -> F ~~> 0 )
6	1, 2, 3, 4, 5	climi 11997	. 2 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < C ) )
7		subid1 8509	. . . . . . 7 |- ( B e. CC -> ( B - 0 ) = B )
8	7	fveq2d 5679	. . . . . 6 |- ( B e. CC -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
9	8	breq1d 4124	. . . . 5 |- ( B e. CC -> ( ( abs ` ( B - 0 ) ) < C <-> ( abs ` B ) < C ) )
10	9	biimpa 296	. . . 4 |- ( ( B e. CC /\ ( abs ` ( B - 0 ) ) < C ) -> ( abs ` B ) < C )
11	10	ralimi 2607	. . 3 |- ( A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < C ) -> A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )
12	11	reximi 2641	. 2 |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - 0 ) ) < C ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )
13	6, 12	syl 14	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   < clt 8324   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   RR+crp 10004   abscabs 11707   ~~> cli 11988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-clim 11989

This theorem is referenced by:  mertenslem2  12247