climisp - Mathbox for Glauco Siliprandi

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Theorem climisp 44452

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.)

**Hypotheses**
Ref	Expression
climisp.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climisp.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climisp.f	⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
climisp.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climisp.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
climisp.l	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))

**Assertion**
Ref	Expression
climisp	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹,𝑘 𝑗,𝑀 𝑗,𝑋,𝑘 𝑗,𝑍,𝑘 𝜑,𝑗,𝑘 Allowed substitution hint: 𝑀(𝑘)

Proof of Theorem climisp Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1917	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
2		nfra1 3281	. . . 4 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
3	1, 2	nfan 1902	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
4		simplll 773	. . . 4 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
5		climisp.z	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6	5	uztrn2 12840	. . . . 5 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
7	6	ad4ant24 752	. . . 4 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
8		rspa 3245	. . . . . 6 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
9	8	simprd 496	. . . . 5 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
10	9	adantll 712	. . . 4 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
11		simpl3 1193	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ∧ ¬ (𝐹‘𝑘) = 𝐴) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
12		neqne 2948	. . . . . . 7 ⊢ (¬ (𝐹‘𝑘) = 𝐴 → (𝐹‘𝑘) ≠ 𝐴)
13		climisp.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
14	13	rpred 13015	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ ℝ)
15	14	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ∈ ℝ)
16		climisp.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
17	16	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
18		climisp.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
19	5	fvexi 6905	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 ∈ V
20	19	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ∈ V)
21	16, 20	fexd 7228	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ V)
22		eqidd 2733	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
23	21, 22	clim 15437	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
24	18, 23	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
25	24	simpld 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℂ)
26	25	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)
27	17, 26	subcld 11570	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
28	27	abscld 15382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
29	28	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
30		climisp.l	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
31	30	3expa 1118	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
32	15, 29, 31	lensymd 11364	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ (𝐹‘𝑘) ≠ 𝐴) → ¬ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
33	12, 32	sylan2 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ ¬ (𝐹‘𝑘) = 𝐴) → ¬ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
34	33	3adantl3 1168	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ∧ ¬ (𝐹‘𝑘) = 𝐴) → ¬ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
35	11, 34	condan 816	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) = 𝐴)
36	4, 7, 10, 35	syl3anc 1371	. . 3 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) = 𝐴)
37	3, 36	ralrimia 3255	. 2 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)
38		breq2 5152	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
39	38	anbi2d 629	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)))
40	39	rexralbidv 3220	. . . 4 ⊢ (𝑥 = 𝑋 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)))
41	24	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
42	40, 41, 13	rspcdva 3613	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
43		climisp.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
44	5	rexuz3 15294	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)))
45	43, 44	syl 17	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)))
46	42, 45	mpbird 256	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
47	37, 46	reximddv3 43830	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝐹‘𝑘) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ℂcc 11107 ℝcr 11108 < clt 11247 ≤ cle 11248 − cmin 11443 ℤcz 12557 ℤ_≥cuz 12821 ℝ⁺crp 12973 abscabs 15180 ⇝ cli 15427

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-seq 13966 df-exp 14027 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-clim 15431

This theorem is referenced by: (None)

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