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Theorem ulmdvlem3 25905

Description: Lemma for ulmdv 25906. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
ulmdv.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
ulmdv.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
ulmdv.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulmdv.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑋))
ulmdv.g	⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
ulmdv.l	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
ulmdv.u	⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)

**Assertion**
Ref	Expression
ulmdvlem3	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))

Distinct variable groups: 𝑧,𝑘,𝐹 𝑧,𝐺 𝑧,𝐻 𝑘,𝑀 𝜑,𝑘,𝑧 𝑆,𝑘,𝑧 𝑘,𝑋,𝑧 𝑘,𝑍,𝑧 Allowed substitution hints: 𝐺(𝑘) 𝐻(𝑘) 𝑀(𝑧)

Proof of Theorem ulmdvlem3 Dummy variables 𝑗 𝑚 𝑛 𝑠 𝑢 𝑣 𝑤 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 261	. . . 4 ⊢ (𝑘 = 𝑀 → (𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ↔ 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋)))
2		ulmdv.z	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3		ulmdv.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
4		ulmdv.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		ulmdv.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑_m 𝑋))
6		ulmdv.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ)
7		ulmdv.l	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
8		ulmdv.u	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
9	2, 3, 4, 5, 6, 7, 8	ulmdvlem2 25904	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
10		recnprss 25412	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑆 ⊆ ℂ)
13	5	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_m 𝑋))
14		elmapi 8839	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ∈ (ℂ ↑_m 𝑋) → (𝐹‘𝑘):𝑋⟶ℂ)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘):𝑋⟶ℂ)
16		dvbsss 25410	. . . . . . . 8 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
17	9, 16	eqsstrrdi 4036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
18		eqid 2732	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
19		eqid 2732	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20	12, 15, 17, 18, 19	dvbssntr 25408	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
21	9, 20	eqsstrrd 4020	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
22	21	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
23		uzid 12833	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
24	4, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
25	24, 2	eleqtrrdi 2844	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
26	1, 22, 25	rspcdva 3613	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
27	26	sselda 3981	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
28		ulmcl 25884	. . . . 5 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻 → 𝐻:𝑋⟶ℂ)
29	8, 28	syl 17	. . . 4 ⊢ (𝜑 → 𝐻:𝑋⟶ℂ)
30	29	ffvelcdmda 7083	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐻‘𝑧) ∈ ℂ)
31		breq2 5151	. . . . . . . 8 ⊢ (𝑠 = ((𝑟 / 2) / 2) → ((abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
32	31	2ralbidv 3218	. . . . . . 7 ⊢ (𝑠 = ((𝑟 / 2) / 2) → (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ ∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
33	32	rexralbidv 3220	. . . . . 6 ⊢ (𝑠 = ((𝑟 / 2) / 2) → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠 ↔ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
34		ulmrel 25881	. . . . . . . . . 10 ⊢ Rel (⇝_𝑢‘𝑋)
35		releldm 5941	. . . . . . . . . 10 ⊢ ((Rel (⇝_𝑢‘𝑋) ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom (⇝_𝑢‘𝑋))
36	34, 8, 35	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom (⇝_𝑢‘𝑋))
37		ulmscl 25882	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻 → 𝑋 ∈ V)
38	8, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		ovex 7438	. . . . . . . . . . . . 13 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V
40	39	rgenw 3065	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V
41		eqid 2732	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))
42	41	fnmpt 6687	. . . . . . . . . . . 12 ⊢ (∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
43	40, 42	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍)
44		ulmf2 25887	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) Fn 𝑍 ∧ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_m 𝑋))
45	43, 8, 44	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_m 𝑋))
46	2, 4, 38, 45	ulmcau2 25899	. . . . . . . . 9 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom (⇝_𝑢‘𝑋) ↔ ∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠))
47	36, 46	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠)
48	2	uztrn2 12837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
49	48	ad2ant2lr 746	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑛 ∈ 𝑍)
50		fveq2 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
51	50	oveq2d 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑛)))
52		ovex 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 D (𝐹‘𝑛)) ∈ V
53	51, 41, 52	fvmpt 6995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
54	49, 53	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
55	54	fveq1d 6890	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) = ((𝑆 D (𝐹‘𝑛))‘𝑥))
56		simprr 771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑚 ∈ (ℤ_≥‘𝑛))
57	2	uztrn2 12837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
58	49, 56, 57	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑚 ∈ 𝑍)
59		fveq2 6888	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
60	59	oveq2d 7421	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑚)))
61		ovex 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 D (𝐹‘𝑚)) ∈ V
62	60, 41, 61	fvmpt 6995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚) = (𝑆 D (𝐹‘𝑚)))
63	58, 62	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚) = (𝑆 D (𝐹‘𝑚)))
64	63	fveq1d 6890	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥) = ((𝑆 D (𝐹‘𝑚))‘𝑥))
65	55, 64	oveq12d 7423	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → ((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥)) = (((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥)))
66	65	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) = (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))))
67	66	breq1d 5157	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → ((abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠))
68	67	ralbidv 3177	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → (∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠))
69	68	2ralbidva 3216	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠))
70	69	rexbidva 3176	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠))
71	70	ralbidv 3177	. . . . . . . 8 ⊢ (𝜑 → (∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘((((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑥) − (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠))
72	47, 71	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)
73	72	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∀𝑠 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < 𝑠)
74		rphalfcl 12997	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
75	74	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 2) ∈ ℝ⁺)
76		rphalfcl 12997	. . . . . . 7 ⊢ ((𝑟 / 2) ∈ ℝ⁺ → ((𝑟 / 2) / 2) ∈ ℝ⁺)
77	75, 76	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ((𝑟 / 2) / 2) ∈ ℝ⁺)
78	33, 73, 77	rspcdva 3613	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))
79	4	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
80	51	fveq1d 6890	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑆 D (𝐹‘𝑘))‘𝑧) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
81		eqid 2732	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))
82		fvex 6901	. . . . . . . 8 ⊢ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ V
83	80, 81, 82	fvmpt 6995	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
84	83	adantl 482	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
85	45	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_m 𝑋))
86		simplr 767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝑧 ∈ 𝑋)
87	2	fvexi 6902	. . . . . . . . 9 ⊢ 𝑍 ∈ V
88	87	mptex 7221	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ∈ V
89	88	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ∈ V)
90	53	adantl 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
91	90	fveq1d 6890	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑧) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
92	91, 84	eqtr4d 2775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛)‘𝑧) = ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛))
93	8	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
94	2, 79, 85, 86, 89, 92, 93	ulmclm 25890	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ⇝ (𝐻‘𝑧))
95	2, 79, 75, 84, 94	climi2 15451	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))
96	2	rexanuz2 15292	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ↔ (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))
97	2	r19.2uz 15294	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))
98	96, 97	sylbir 234	. . . . . 6 ⊢ ((∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))
99		fveq2 6888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑛)‘𝑣))
100	99	oveq1d 7420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) = (((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)))
101		oveq1 7412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (𝑦 − 𝑧) = (𝑣 − 𝑧))
102	100, 101	oveq12d 7423	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) = ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)))
103		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))
104		ovex 7438	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) ∈ V
105	102, 103, 104	fvmpt 6995	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) = ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)))
106	105	fvoveq1d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) = (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))))
107		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ((𝑟 / 2) / 2) → 𝑠 = ((𝑟 / 2) / 2))
108	106, 107	breqan12rd 5164	. . . . . . . . . . . . 13 ⊢ ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠 ↔ (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
109	108	imbi2d 340	. . . . . . . . . . . 12 ⊢ ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
110	109	ralbidva 3175	. . . . . . . . . . 11 ⊢ (𝑠 = ((𝑟 / 2) / 2) → (∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
111	110	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑠 = ((𝑟 / 2) / 2) → (∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠) ↔ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
112		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ 𝑋)
113	85	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) ∈ (ℂ ↑_m 𝑋))
114	90, 113	eqeltrrd 2834	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_m 𝑋))
115		elmapi 8839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_m 𝑋) → (𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ)
116		fdm 6723	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ → dom (𝑆 D (𝐹‘𝑛)) = 𝑋)
117	114, 115, 116	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑛)) = 𝑋)
118	112, 117	eleqtrrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)))
119	3	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ {ℝ, ℂ})
120		dvfg 25414	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ)
121		ffun 6717	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ → Fun (𝑆 D (𝐹‘𝑛)))
122		funfvbrb 7049	. . . . . . . . . . . . . . . 16 ⊢ (Fun (𝑆 D (𝐹‘𝑛)) → (𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)) ↔ 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧)))
123	119, 120, 121, 122	4syl 19	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)) ↔ 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧)))
124	118, 123	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧))
125	119, 10	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑆 ⊆ ℂ)
126	5	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_m 𝑋))
127	126	ffvelcdmda 7083	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑_m 𝑋))
128		elmapi 8839	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑛) ∈ (ℂ ↑_m 𝑋) → (𝐹‘𝑛):𝑋⟶ℂ)
129	127, 128	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):𝑋⟶ℂ)
130		biidd 261	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑀 → (𝑋 ⊆ 𝑆 ↔ 𝑋 ⊆ 𝑆))
131	17	ralrimiva 3146	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
132	130, 131, 25	rspcdva 3613	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
133	132	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
134	18, 19, 103, 125, 129, 133	eldv 25406	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑧(𝑆 D (𝐹‘𝑛))((𝑆 D (𝐹‘𝑛))‘𝑧) ↔ (𝑧 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧))))
135	124, 134	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑧 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧)))
136	135	simprd 496	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧))
137	132	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ 𝑆)
138	11	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑆 ⊆ ℂ)
139	137, 138	sstrd 3991	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ ℂ)
140	139	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ ℂ)
141	129, 140, 112	dvlem 25404	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ)
142	141	fmpttd 7111	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))):(𝑋 ∖ {𝑧})⟶ℂ)
143	140	ssdifssd 4141	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
144	140, 112	sseldd 3982	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ ℂ)
145	142, 143, 144	ellimc3 25387	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧) ↔ (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠))))
146	136, 145	mpbid 231	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠)))
147	146	simprd 496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ∀𝑠 ∈ ℝ⁺ ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < 𝑠))
148	77	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑟 / 2) / 2) ∈ ℝ⁺)
149	111, 147, 148	rspcdva 3613	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
150	149	adantrr 715	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) → ∃𝑤 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
151		anass 469	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) ↔ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺)))
152		df-3an 1089	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))) ↔ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))
153		anass 469	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)))
154	7	ralrimiva 3146	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
155		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑠 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑠))
156	155	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑠 → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)))
157		fveq2 6888	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑠 → (𝐺‘𝑧) = (𝐺‘𝑠))
158	156, 157	breq12d 5160	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑠 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ↔ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠)))
159	158	rspccva 3611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠))
160	154, 159	sylan 580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠))
161		simprll 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑧 ∈ 𝑋)
162		simprlr 778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑟 ∈ ℝ⁺)
163		simprr3 1223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))
164		simplll 773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑢 ∈ ℝ⁺)
165	163, 164	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑢 ∈ ℝ⁺)
166		simplr 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑤 ∈ ℝ⁺)
167	163, 166	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑤 ∈ ℝ⁺)
168		simpllr 774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
169	163, 168	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
170	169	simpld 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑢 < 𝑤)
171	169	simprd 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)
172		simpr3 1196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))
173	163, 172	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))
174	173	simprd 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘(𝑣 − 𝑧)) < 𝑢)
175		simprr1 1221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑛 ∈ 𝑍)
176		simprr2 1222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))
177	176	simpld 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2))
178	176	simprd 496	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))
179		simpr1 1194	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → 𝑣 ∈ (𝑋 ∖ {𝑧}))
180	163, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ∈ (𝑋 ∖ {𝑧}))
181	180	eldifad 3959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ∈ 𝑋)
182	173	simpld 495	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → 𝑣 ≠ 𝑧)
183		simpr2 1195	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
184	163, 183	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
185	182, 184	mpand 693	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → ((abs‘(𝑣 − 𝑧)) < 𝑤 → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
186	2, 3, 4, 5, 6, 160, 8, 161, 162, 165, 167, 170, 171, 174, 175, 177, 178, 181, 182, 185	ulmdvlem1 25903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
187	186	anassrs 468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
188	153, 187	sylanb 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
189	152, 188	sylan2br 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ ((𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
190	189	anassrs 468	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢)))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
191	190	anassrs 468	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ⁺)) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
192	151, 191	sylanb 581	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢))) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)
193	192	3exp2 1354	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) → (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟))))
194	193	imp 407	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)))
195		fveq2 6888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → (𝐺‘𝑦) = (𝐺‘𝑣))
196	195	oveq1d 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → ((𝐺‘𝑦) − (𝐺‘𝑧)) = ((𝐺‘𝑣) − (𝐺‘𝑧)))
197	196, 101	oveq12d 7423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)))
198		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))
199		ovex 7438	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) ∈ V
200	197, 198, 199	fvmpt 6995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)))
201	200	fvoveq1d 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) = (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))))
202	201	breq1d 5157	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟 ↔ (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟))
203	202	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)))
204	203	adantl 482	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟) ↔ ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘((((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) − (𝐻‘𝑧))) < 𝑟)))
205	194, 204	sylibrd 258	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)))
206	205	ralimdva 3167	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ⁺) → (∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)))
207	206	impr 455	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
208	207	an32s 650	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) ∧ (𝑢 ∈ ℝ⁺ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
209		cnxmet 24280	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
210		xmetres2 23858	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
211	209, 138, 210	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
212	211	ad3antrrr 728	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
213	19	cnfldtop 24291	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ Top
214		resttop 22655	. . . . . . . . . . . . . . . . 17 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ {ℝ, ℂ}) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
215	213, 3, 214	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
216	19	cnfldtopon 24290	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
217		resttopon 22656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
218	216, 11, 217	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
219		toponuni 22407	. . . . . . . . . . . . . . . . . 18 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
220	218, 219	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
221	132, 220	sseqtrd 4021	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
222		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
223	222	ntrss2 22552	. . . . . . . . . . . . . . . 16 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)) → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ⊆ 𝑋)
224	215, 221, 223	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ⊆ 𝑋)
225	224, 26	eqssd 3998	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) = 𝑋)
226	222	isopn3 22561	. . . . . . . . . . . . . . 15 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ 𝑋 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)) → (𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ↔ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) = 𝑋))
227	215, 221, 226	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ↔ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) = 𝑋))
228	225, 227	mpbird 256	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
229		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
230	19	cnfldtopn 24289	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
231		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
232	229, 230, 231	metrest 24024	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
233	209, 11, 232	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
234	228, 233	eleqtrd 2835	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
235	234	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
236	235	ad3antrrr 728	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
237	86	ad2antrr 724	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑧 ∈ 𝑋)
238		simprl 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑤 ∈ ℝ⁺)
239	231	mopni3 23994	. . . . . . . . . 10 ⊢ (((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑤 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
240	212, 236, 237, 238, 239	syl31anc 1373	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ⁺ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
241	208, 240	reximddv 3171	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ⁺ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑤) → (abs‘(((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) − ((𝑆 D (𝐹‘𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
242	150, 241	rexlimddv 3161	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑛 ∈ 𝑍 ∧ (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)))) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
243	242	rexlimdvaa 3156	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (∃𝑛 ∈ 𝑍 (∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)))
244	98, 243	syl5 34	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ((∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)∀𝑚 ∈ (ℤ_≥‘𝑛)∀𝑥 ∈ 𝑋 (abs‘(((𝑆 D (𝐹‘𝑛))‘𝑥) − ((𝑆 D (𝐹‘𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟)))
245	78, 95, 244	mp2and 697	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
246	245	ralrimiva 3146	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑟 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))
247	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐺:𝑋⟶ℂ)
248		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
249	247, 139, 248	dvlem 25404	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ)
250	249	fmpttd 7111	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))):(𝑋 ∖ {𝑧})⟶ℂ)
251	139	ssdifssd 4141	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
252	139, 248	sseldd 3982	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ℂ)
253	250, 251, 252	ellimc3 25387	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧) ↔ ((𝐻‘𝑧) ∈ ℂ ∧ ∀𝑟 ∈ ℝ⁺ ∃𝑢 ∈ ℝ⁺ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣 ≠ 𝑧 ∧ (abs‘(𝑣 − 𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) − (𝐻‘𝑧))) < 𝑟))))
254	30, 246, 253	mpbir2and 711	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧))
255	18, 19, 198, 138, 247, 137	eldv 25406	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑧(𝑆 D 𝐺)(𝐻‘𝑧) ↔ (𝑧 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ (𝐻‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) lim_ℂ 𝑧))))
256	27, 254, 255	mpbir2and 711	1 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 {cpr 4629 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 dom cdm 5675 ↾ cres 5677 ∘ ccom 5679 Rel wrel 5680 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ↑_m cmap 8816 ℂcc 11104 ℝcr 11105 < clt 11244 − cmin 11440 / cdiv 11867 2c2 12263 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 abscabs 15177 ⇝ cli 15424 ↾_t crest 17362 TopOpenctopn 17363 ∞Metcxmet 20921 ballcbl 20923 MetOpencmopn 20926 ℂ_fldccnfld 20936 Topctop 22386 TopOnctopon 22403 intcnt 22512 lim_ℂ climc 25370 D cdv 25371 ⇝_𝑢culm 25879

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-ulm 25880

This theorem is referenced by: ulmdv 25906

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