Metamath Proof Explorer


Theorem climisp

Description: If a sequence converges to an isolated point (w.r.t. the standard topology on the complex numbers) then the sequence eventually becomes that point. (Contributed by Glauco Siliprandi, 5-Feb-2022)

Ref Expression
Hypotheses climisp.m φ M
climisp.z Z = M
climisp.f φ F : Z
climisp.c φ F A
climisp.x φ X +
climisp.l φ k Z F k A X F k A
Assertion climisp φ j Z k j F k = A

Proof

Step Hyp Ref Expression
1 climisp.m φ M
2 climisp.z Z = M
3 climisp.f φ F : Z
4 climisp.c φ F A
5 climisp.x φ X +
6 climisp.l φ k Z F k A X F k A
7 nfv k φ j Z
8 nfra1 k k j F k F k A < X
9 7 8 nfan k φ j Z k j F k F k A < X
10 simplll φ j Z k j F k F k A < X k j φ
11 2 uztrn2 j Z k j k Z
12 11 ad4ant24 φ j Z k j F k F k A < X k j k Z
13 rspa k j F k F k A < X k j F k F k A < X
14 13 simprd k j F k F k A < X k j F k A < X
15 14 adantll φ j Z k j F k F k A < X k j F k A < X
16 simpl3 φ k Z F k A < X ¬ F k = A F k A < X
17 neqne ¬ F k = A F k A
18 5 rpred φ X
19 18 ad2antrr φ k Z F k A X
20 3 ffvelrnda φ k Z F k
21 2 fvexi Z V
22 21 a1i φ Z V
23 3 22 fexd φ F V
24 eqidd φ k F k = F k
25 23 24 clim φ F A A x + j k j F k F k A < x
26 4 25 mpbid φ A x + j k j F k F k A < x
27 26 simpld φ A
28 27 adantr φ k Z A
29 20 28 subcld φ k Z F k A
30 29 abscld φ k Z F k A
31 30 adantr φ k Z F k A F k A
32 6 3expa φ k Z F k A X F k A
33 19 31 32 lensymd φ k Z F k A ¬ F k A < X
34 17 33 sylan2 φ k Z ¬ F k = A ¬ F k A < X
35 34 3adantl3 φ k Z F k A < X ¬ F k = A ¬ F k A < X
36 16 35 condan φ k Z F k A < X F k = A
37 10 12 15 36 syl3anc φ j Z k j F k F k A < X k j F k = A
38 9 37 ralrimia φ j Z k j F k F k A < X k j F k = A
39 breq2 x = X F k A < x F k A < X
40 39 anbi2d x = X F k F k A < x F k F k A < X
41 40 rexralbidv x = X j k j F k F k A < x j k j F k F k A < X
42 26 simprd φ x + j k j F k F k A < x
43 41 42 5 rspcdva φ j k j F k F k A < X
44 2 rexuz3 M j Z k j F k F k A < X j k j F k F k A < X
45 1 44 syl φ j Z k j F k F k A < X j k j F k F k A < X
46 43 45 mpbird φ j Z k j F k F k A < X
47 38 46 reximddv3 φ j Z k j F k = A