Theorem climuz 39976

Description: Express the predicate: The limit of complex number sequence F is A, or F converges to A. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climuz.k	|- F/_ k F
climuz.m	|- ( ph -> M e. ZZ )
climuz.z	|- Z = ( ZZ>= ` M )
climuz.f	|- ( ph -> F : Z --> CC )

Assertion

Ref	Expression
climuz	|- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) ) )

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

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

Proof of Theorem climuz

Dummy variables i l y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climuz.m	. . 3 |- ( ph -> M e. ZZ )
2		climuz.z	. . 3 |- Z = ( ZZ>= ` M )
3		climuz.f	. . 3 |- ( ph -> F : Z --> CC )
4	1, 2, 3	climuzlem 39975	. 2 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. y e. RR+ E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y ) ) )
5		breq2 4657	. . . . . . . 8 |- ( y = x -> ( ( abs ` ( ( F ` l ) - A ) ) < y <-> ( abs ` ( ( F ` l ) - A ) ) < x ) )
6	5	ralbidv 2986	. . . . . . 7 |- ( y = x -> ( A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y <-> A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x ) )
7	6	rexbidv 3052	. . . . . 6 |- ( y = x -> ( E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y <-> E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x ) )
8		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( ZZ>= ` i ) = ( ZZ>= ` j ) )
9	8	raleqdv 3144	. . . . . . . . 9 |- ( i = j -> ( A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x <-> A. l e. ( ZZ>= ` j ) ( abs ` ( ( F ` l ) - A ) ) < x ) )
10		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ k abs
11		climuz.k	. . . . . . . . . . . . . . 15 |- F/_ k F
12		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ k l
13	11, 12	nffv 6198	. . . . . . . . . . . . . 14 |- F/_ k ( F ` l )
14		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ k -
15		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ k A
16	13, 14, 15	nfov 6676	. . . . . . . . . . . . 13 |- F/_ k ( ( F ` l ) - A )
17	10, 16	nffv 6198	. . . . . . . . . . . 12 |- F/_ k ( abs ` ( ( F ` l ) - A ) )
18		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k <
19		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k x
20	17, 18, 19	nfbr 4699	. . . . . . . . . . 11 |- F/ k ( abs ` ( ( F ` l ) - A ) ) < x
21		nfv 1843	. . . . . . . . . . 11 |- F/ l ( abs ` ( ( F ` k ) - A ) ) < x
22		fveq2 6191	. . . . . . . . . . . . . 14 |- ( l = k -> ( F ` l ) = ( F ` k ) )
23	22	oveq1d 6665	. . . . . . . . . . . . 13 |- ( l = k -> ( ( F ` l ) - A ) = ( ( F ` k ) - A ) )
24	23	fveq2d 6195	. . . . . . . . . . . 12 |- ( l = k -> ( abs ` ( ( F ` l ) - A ) ) = ( abs ` ( ( F ` k ) - A ) ) )
25	24	breq1d 4663	. . . . . . . . . . 11 |- ( l = k -> ( ( abs ` ( ( F ` l ) - A ) ) < x <-> ( abs ` ( ( F ` k ) - A ) ) < x ) )
26	20, 21, 25	cbvral 3167	. . . . . . . . . 10 |- ( A. l e. ( ZZ>= ` j ) ( abs ` ( ( F ` l ) - A ) ) < x <-> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
27	26	a1i 11	. . . . . . . . 9 |- ( i = j -> ( A. l e. ( ZZ>= ` j ) ( abs ` ( ( F ` l ) - A ) ) < x <-> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) )
28	9, 27	bitrd 268	. . . . . . . 8 |- ( i = j -> ( A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x <-> A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) )
29	28	cbvrexv 3172	. . . . . . 7 |- ( E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
30	29	a1i 11	. . . . . 6 |- ( y = x -> ( E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) )
31	7, 30	bitrd 268	. . . . 5 |- ( y = x -> ( E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) )
32	31	cbvralv 3171	. . . 4 |- ( A. y e. RR+ E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x )
33	32	anbi2i 730	. . 3 |- ( ( A e. CC /\ A. y e. RR+ E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y ) <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) )
34	33	a1i 11	. 2 |- ( ph -> ( ( A e. CC /\ A. y e. RR+ E. i e. Z A. l e. ( ZZ>= ` i ) ( abs ` ( ( F ` l ) - A ) ) < y ) <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
35	4, 34	bitrd 268	1 |- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - A ) ) < x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E.wrex 2913   class class class wbr 4653   -->wf 5884   ` 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-rep 4771  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-reu 2919  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-iun 4522  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:  liminflimsupclim  40039  climxlim2lem  40071