Theorem climd 39904

Description: Express the predicate: The limit of complex number sequence F is A, or F converges to A. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
climd.1	|- F/ k ph
climd.2	|- F/_ k F
climd.3	|- Z = ( ZZ>= ` M )
climd.4	|- ( ph -> M e. ZZ )
climd.5	|- ( ph -> F ~~> A )
climd.6	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
climd.7	|- ( ph -> X e. RR+ )

Assertion

Ref	Expression
climd	|- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < X ) )

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

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

Proof of Theorem climd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climd.7	. 2 |- ( ph -> X e. RR+ )
2		climd.5	. . . 4 |- ( ph -> F ~~> A )
3		climd.1	. . . . 5 |- F/ k ph
4		climd.2	. . . . 5 |- F/_ k F
5		climd.3	. . . . 5 |- Z = ( ZZ>= ` M )
6		climd.4	. . . . 5 |- ( ph -> M e. ZZ )
7		climrel 14223	. . . . . . 7 |- Rel ~~>
8	7	brrelexi 5158	. . . . . 6 |- ( F ~~> A -> F e. _V )
9	2, 8	syl 17	. . . . 5 |- ( ph -> F e. _V )
10		climd.6	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
11	3, 4, 5, 6, 9, 10	clim2f2 39902	. . . 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	2, 11	mpbid 222	. . 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 479	. 2 |- ( ph -> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) )
14		breq2 4657	. . . . 5 |- ( x = X -> ( ( abs ` ( B - A ) ) < x <-> ( abs ` ( B - A ) ) < X ) )
15	14	anbi2d 740	. . . 4 |- ( x = X -> ( ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < X ) ) )
16	15	rexralbidv 3058	. . 3 |- ( x = X -> ( 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 ) ) < X ) ) )
17	16	rspcva 3307	. 2 |- ( ( X e. RR+ /\ A. x e. RR+ 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 ) ) < X ) )
18	1, 13, 17	syl2anc 693	1 |- ( ph -> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-clim 14219

This theorem is referenced by:  fnlimabslt  39911