Theorem rlim 15516

Description: Express the predicate: The limit of complex number function 𝐹 is 𝐶, or 𝐹 converges to 𝐶, in the real sense. This means that for any real 𝑥, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than 𝑥. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
rlim.1	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
rlim.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
rlim.4	⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)

Assertion

Ref	Expression
rlim	⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))))

Distinct variable groups:   𝑧,𝐴   𝑥,𝑦,𝑧,𝐶   𝑥,𝐹,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑧)

Proof of Theorem rlim

Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlimrel 15514	. . . . 5 ⊢ Rel ⇝𝑟
2	1	brrelex2i 5716	. . . 4 ⊢ (𝐹 ⇝𝑟 𝐶 → 𝐶 ∈ V)
3	2	a1i 11	. . 3 ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 → 𝐶 ∈ V))
4		elex 3485	. . . . 5 ⊢ (𝐶 ∈ ℂ → 𝐶 ∈ V)
5	4	ad2antrl 728	. . . 4 ⊢ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V)
6	5	a1i 11	. . 3 ⊢ (𝜑 → ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))) → 𝐶 ∈ V))
7		rlim.1	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
8		rlim.2	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9		cnex 11215	. . . . . 6 ⊢ ℂ ∈ V
10		reex 11225	. . . . . 6 ⊢ ℝ ∈ V
11		elpm2r 8864	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ))
12	9, 10, 11	mpanl12 702	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ))
13	7, 8, 12	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℝ))
14		eleq1 2823	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ (ℂ ↑pm ℝ) ↔ 𝐹 ∈ (ℂ ↑pm ℝ)))
15		eleq1 2823	. . . . . . . . 9 ⊢ (𝑤 = 𝐶 → (𝑤 ∈ ℂ ↔ 𝐶 ∈ ℂ))
16	14, 15	bi2anan9 638	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ)))
17		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → 𝑓 = 𝐹)
18	17	dmeqd 5890	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → dom 𝑓 = dom 𝐹)
19		fveq1 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
20		oveq12 7419	. . . . . . . . . . . . . . 15 ⊢ (((𝑓‘𝑧) = (𝐹‘𝑧) ∧ 𝑤 = 𝐶) → ((𝑓‘𝑧) − 𝑤) = ((𝐹‘𝑧) − 𝐶))
21	19, 20	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑓‘𝑧) − 𝑤) = ((𝐹‘𝑧) − 𝐶))
22	21	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (abs‘((𝑓‘𝑧) − 𝑤)) = (abs‘((𝐹‘𝑧) − 𝐶)))
23	22	breq1d 5134	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥 ↔ (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))
24	23	imbi2d 340	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → ((𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
25	18, 24	raleqbidv 3329	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
26	25	rexbidv 3165	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
27	26	ralbidv 3164	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
28	16, 27	anbi12d 632	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑤 = 𝐶) → (((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥)) ↔ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
29		df-rlim 15510	. . . . . . 7 ⊢ ⇝𝑟 = {⟨𝑓, 𝑤⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑤 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝑓(𝑦 ≤ 𝑧 → (abs‘((𝑓‘𝑧) − 𝑤)) < 𝑥))}
30	28, 29	brabga 5514	. . . . . 6 ⊢ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹 ⇝𝑟 𝐶 ↔ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
31		anass 468	. . . . . 6 ⊢ (((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))
32	30, 31	bitrdi 287	. . . . 5 ⊢ ((𝐹 ∈ (ℂ ↑pm ℝ) ∧ 𝐶 ∈ V) → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
33	32	ex 412	. . . 4 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → (𝐶 ∈ V → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))))
34	13, 33	syl 17	. . 3 ⊢ (𝜑 → (𝐶 ∈ V → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥))))))
35	3, 6, 34	pm5.21ndd 379	. 2 ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
36	13	biantrurd 532	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))))
37	7	fdmd 6721	. . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴)
38	37	raleqdv 3309	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)))
39		rlim.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)
40	39	fvoveq1d 7432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (abs‘((𝐹‘𝑧) − 𝐶)) = (abs‘(𝐵 − 𝐶)))
41	40	breq1d 5134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑥))
42	41	imbi2d 340	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
43	42	ralbidva 3162	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
44	38, 43	bitrd 279	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
45	44	rexbidv 3165	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
46	45	ralbidv 3164	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
47	46	anbi2d 630	. 2 ⊢ (𝜑 → ((𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ dom 𝐹(𝑦 ≤ 𝑧 → (abs‘((𝐹‘𝑧) − 𝐶)) < 𝑥)) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))))
48	35, 36, 47	3bitr2d 307	1 ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124  dom cdm 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8846  ℂcc 11132  ℝcr 11133   < clt 11274   ≤ cle 11275   − cmin 11471  ℝ⁺crp 13013  abscabs 15258   ⇝_𝑟 crli 15506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-pm 8848  df-rlim 15510

This theorem is referenced by:  rlim2  15517  rlimcl  15524  rlimclim  15567  rlimres  15579  caurcvgr  15695