Theorem clim2 9804

Description: Express the predicate: The limit of complex number sequence F is A, or F converges to A, with more general quantifier restrictions than clim 9802. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)

Hypotheses

Ref	Expression
clim2.1	|- Z = ( ZZ>= ` M )
clim2.2	|- ( ph -> M e. ZZ )
clim2.3	|- ( ph -> F e. V )
clim2.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )

Assertion

Ref	Expression
clim2	|- ( 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 ) ) ) )

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

Allowed substitution hints:   B( x, j, k)   M( x, k)   V( x, j, k)   Z( x)

Proof of Theorem clim2

Step	Hyp	Ref	Expression
1		clim2.3	. . 3 |- ( ph -> F e. V )
2		eqidd 2041	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
3	1, 2	clim 9802	. 2 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
4		clim2.1	. . . . . . . . . 10 |- Z = ( ZZ>= ` M )
5	4	uztrn2 8490	. . . . . . . . 9 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
6		clim2.4	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
7	6	eleq1d 2106	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) e. CC <-> B e. CC ) )
8	6	oveq1d 5527	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) - A ) = ( B - A ) )
9	8	fveq2d 5182	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( abs ` ( B - A ) ) )
10	9	breq1d 3774	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x <-> ( abs ` ( B - A ) ) < x ) )
11	7, 10	anbi12d 442	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
12	5, 11	sylan2 270	. . . . . . . 8 |- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
13	12	anassrs 380	. . . . . . 7 |- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
14	13	ralbidva 2322	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
15	14	rexbidva 2323	. . . . 5 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
16		clim2.2	. . . . . 6 |- ( ph -> M e. ZZ )
17	4	rexuz3 9588	. . . . . 6 |- ( M e. ZZ -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
18	16, 17	syl 14	. . . . 5 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
19	15, 18	bitr3d 179	. . . 4 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
20	19	ralbidv 2326	. . 3 |- ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
21	20	anbi2d 437	. 2 |- ( ph -> ( ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
22	3, 21	bitr4d 180	1 |- ( 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 ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   CCcc 6887   < clt 7060   - cmin 7182   ZZcz 8245   ZZ>=cuz 8473   RR+crp 8583   abscabs 9595   ~~> cli 9799

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-clim 9800

This theorem is referenced by:  clim2c  9805  clim0  9806  climi  9808  climeq  9820