Theorem clim2 12074

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 12072. (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 2239	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = ( F ` k ) )
3	1, 2	clim 12072	. 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 9956	. . . . . . . . 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 2307	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) e. CC <-> B e. CC ) )
8	6	oveq1d 6103	. . . . . . . . . . . 12 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) - A ) = ( B - A ) )
9	8	fveq2d 5702	. . . . . . . . . . 11 |- ( ( ph /\ k e. Z ) -> ( abs ` ( ( F ` k ) - A ) ) = ( abs ` ( B - A ) ) )
10	9	breq1d 4141	. . . . . . . . . 10 |- ( ( ph /\ k e. Z ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x <-> ( abs ` ( B - A ) ) < x ) )
11	7, 10	anbi12d 477	. . . . . . . . 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 286	. . . . . . . 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 404	. . . . . . 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 2546	. . . . . 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 2547	. . . . 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 11778	. . . . . 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 190	. . . 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 2550	. . 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 468	. 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 191	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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   < clt 8366   - cmin 8504   ZZcz 9655   ZZ>=cuz 9937   RR+crp 10071   abscabs 11785   ~~> cli 12069

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-clim 12070

This theorem is used by:  clim2c  12075  clim0  12076  climi  12078  climeq  12090