ulmdvlem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ulmdvlem3		Structured version Visualization version GIF version

Theorem ulmdvlem3 24696

Description: Lemma for ulmdv 24697. (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 254	. . . 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 24695	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) = 𝑋)
10		recnprss 24208	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	3, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
12	11	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑆 ⊆ ℂ)
13	5	ffvelrnda 6678	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ (ℂ ↑_𝑚 𝑋))
14		elmapi 8230	. . . . . . . 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 24206	. . . . . . . 8 ⊢ dom (𝑆 D (𝐹‘𝑘)) ⊆ 𝑆
17	9, 16	syl6eqssr 3914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
18		eqid 2778	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
19		eqid 2778	. . . . . . 7 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20	12, 15, 17, 18, 19	dvbssntr 24204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → dom (𝑆 D (𝐹‘𝑘)) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
21	9, 20	eqsstr3d 3898	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
22	21	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
23		uzid 12076	. . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
24	4, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
25	24, 2	syl6eleqr 2877	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
26	1, 22, 25	rspcdva 3541	. . 3 ⊢ (𝜑 → 𝑋 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
27	26	sselda 3860	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋))
28		ulmcl 24675	. . . . 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	ffvelrnda 6678	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝐻‘𝑧) ∈ ℂ)
31		breq2 4934	. . . . . . . 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 3149	. . . . . . 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 3246	. . . . . 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 24672	. . . . . . . . . 10 ⊢ Rel (⇝_𝑢‘𝑋)
35		releldm 5658	. . . . . . . . . 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 578	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) ∈ dom (⇝_𝑢‘𝑋))
37		ulmscl 24673	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻 → 𝑋 ∈ V)
38	8, 37	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ V)
39		ovex 7010	. . . . . . . . . . . . 13 ⊢ (𝑆 D (𝐹‘𝑘)) ∈ V
40	39	rgenw 3100	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑍 (𝑆 D (𝐹‘𝑘)) ∈ V
41		eqid 2778	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))) = (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))
42	41	fnmpt 6320	. . . . . . . . . . . 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 24678	. . . . . . . . . . 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 576	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
46	2, 4, 38, 45	ulmcau2 24690	. . . . . . . . 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 224	. . . . . . . 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 12079	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ_≥‘𝑗)) → 𝑛 ∈ 𝑍)
49	48	ad2ant2lr 735	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑛 ∈ 𝑍)
50		fveq2 6501	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
51	50	oveq2d 6994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑛 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑛)))
52		ovex 7010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 D (𝐹‘𝑛)) ∈ V
53	51, 41, 52	fvmpt 6597	. . . . . . . . . . . . . . . . 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 6503	. . . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑚 ∈ (ℤ_≥‘𝑛))
57	2	uztrn2 12079	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
58	49, 56, 57	syl2anc 576	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝑛 ∈ (ℤ_≥‘𝑗) ∧ 𝑚 ∈ (ℤ_≥‘𝑛))) → 𝑚 ∈ 𝑍)
59		fveq2 6501	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚))
60	59	oveq2d 6994	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → (𝑆 D (𝐹‘𝑘)) = (𝑆 D (𝐹‘𝑚)))
61		ovex 7010	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 D (𝐹‘𝑚)) ∈ V
62	60, 41, 61	fvmpt 6597	. . . . . . . . . . . . . . . . 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 6503	. . . . . . . . . . . . . . 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 6996	. . . . . . . . . . . . . 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 6505	. . . . . . . . . . . . 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 4940	. . . . . . . . . . . 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 3147	. . . . . . . . . . 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 3148	. . . . . . . . . 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 3241	. . . . . . . . 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 3147	. . . . . . . 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 224	. . . . . . 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 713	. . . . . 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 12236	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (𝑟 / 2) ∈ ℝ⁺)
75	74	adantl 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑟 / 2) ∈ ℝ⁺)
76		rphalfcl 12236	. . . . . . 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 3541	. . . . 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 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
80	51	fveq1d 6503	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → ((𝑆 D (𝐹‘𝑘))‘𝑧) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
81		eqid 2778	. . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))
82		fvex 6514	. . . . . . . 8 ⊢ ((𝑆 D (𝐹‘𝑛))‘𝑧) ∈ V
83	80, 81, 82	fvmpt 6597	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
84	83	adantl 474	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹‘𝑛))‘𝑧))
85	45	ad2antrr 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘))):𝑍⟶(ℂ ↑_𝑚 𝑋))
86		simplr 756	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝑧 ∈ 𝑋)
87	2	fvexi 6515	. . . . . . . . 9 ⊢ 𝑍 ∈ V
88	87	mptex 6814	. . . . . . . 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 474	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) = (𝑆 D (𝐹‘𝑛)))
91	90	fveq1d 6503	. . . . . . . 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 2817	. . . . . . 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 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))(⇝_𝑢‘𝑋)𝐻)
94	2, 79, 85, 86, 89, 92, 93	ulmclm 24681	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → (𝑘 ∈ 𝑍 ↦ ((𝑆 D (𝐹‘𝑘))‘𝑧)) ⇝ (𝐻‘𝑧))
95	2, 79, 75, 84, 94	climi2 14732	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ_≥‘𝑗)(abs‘(((𝑆 D (𝐹‘𝑛))‘𝑧) − (𝐻‘𝑧))) < (𝑟 / 2))
96	2	rexanuz2 14573	. . . . . . 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 14575	. . . . . . 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 227	. . . . . 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 6501	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → ((𝐹‘𝑛)‘𝑦) = ((𝐹‘𝑛)‘𝑣))
100	99	oveq1d 6993	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) = (((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)))
101		oveq1 6985	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑣 → (𝑦 − 𝑧) = (𝑣 − 𝑧))
102	100, 101	oveq12d 6996	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) = ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)))
103		eqid 2778	. . . . . . . . . . . . . . . 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 7010	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹‘𝑛)‘𝑣) − ((𝐹‘𝑛)‘𝑧)) / (𝑣 − 𝑧)) ∈ V
105	102, 103, 104	fvmpt 6597	. . . . . . . . . . . . . . 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 7000	. . . . . . . . . . . . . 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 4947	. . . . . . . . . . . . 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 333	. . . . . . . . . . . 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 3146	. . . . . . . . . . 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 3242	. . . . . . . . . 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 763	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ 𝑋)
113	85	ffvelrnda 6678	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ (𝑆 D (𝐹‘𝑘)))‘𝑛) ∈ (ℂ ↑_𝑚 𝑋))
114	90, 113	eqeltrrd 2867	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_𝑚 𝑋))
115		elmapi 8230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 D (𝐹‘𝑛)) ∈ (ℂ ↑_𝑚 𝑋) → (𝑆 D (𝐹‘𝑛)):𝑋⟶ℂ)
116		fdm 6354	. . . . . . . . . . . . . . . . 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 2869	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ dom (𝑆 D (𝐹‘𝑛)))
119	3	ad3antrrr 717	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ {ℝ, ℂ})
120		dvfg 24210	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ)
121		ffun 6349	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 D (𝐹‘𝑛)):dom (𝑆 D (𝐹‘𝑛))⟶ℂ → Fun (𝑆 D (𝐹‘𝑛)))
122		funfvbrb 6648	. . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) → 𝐹:𝑍⟶(ℂ ↑_𝑚 𝑋))
127	126	ffvelrnda 6678	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (ℂ ↑_𝑚 𝑋))
128		elmapi 8230	. . . . . . . . . . . . . . . 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 254	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑀 → (𝑋 ⊆ 𝑆 ↔ 𝑋 ⊆ 𝑆))
131	17	ralrimiva 3132	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 𝑋 ⊆ 𝑆)
132	130, 131, 25	rspcdva 3541	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
133	132	ad3antrrr 717	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ 𝑆)
134	18, 19, 103, 125, 129, 133	eldv 24202	. . . . . . . . . . . . . 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 224	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ 𝑆)
138	11	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑆 ⊆ ℂ)
139	137, 138	sstrd 3870	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ⊆ ℂ)
140	139	ad2antrr 713	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑋 ⊆ ℂ)
141	129, 140, 112	dvlem 24200	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → ((((𝐹‘𝑛)‘𝑦) − ((𝐹‘𝑛)‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ)
142	141	fmpttd 6704	. . . . . . . . . . . . 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 4011	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
144	140, 112	sseldd 3861	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → 𝑧 ∈ ℂ)
145	142, 143, 144	ellimc3 24183	. . . . . . . . . . . 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 224	. . . . . . . . . . 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 488	. . . . . . . . . 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 473	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑛 ∈ 𝑍) → ((𝑟 / 2) / 2) ∈ ℝ⁺)
149	111, 147, 148	rspcdva 3541	. . . . . . . . 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 704	. . . . . . . 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 461	. . . . . . . . . . . . . . . 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 1070	. . . . . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑟 ∈ ℝ⁺) ↔ (𝜑 ∧ (𝑧 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺)))
154	7	ralrimiva 3132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧))
155		fveq2 6501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧 = 𝑠 → ((𝐹‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑠))
156	155	mpteq2dv 5024	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑠 → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) = (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)))
157		fveq2 6501	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = 𝑠 → (𝐺‘𝑧) = (𝐺‘𝑠))
158	156, 157	breq12d 4943	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 = 𝑠 → ((𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ↔ (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠)))
159	158	rspccva 3534	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑧 ∈ 𝑋 (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑧)) ⇝ (𝐺‘𝑧) ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠))
160	154, 159	sylan 572	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑘 ∈ 𝑍 ↦ ((𝐹‘𝑘)‘𝑠)) ⇝ (𝐺‘𝑠))
161		simprll 766	. . . . . . . . . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . 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 1176	. . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . 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 1201	. . . . . . . . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . 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 1174	. . . . . . . . . . . . . . . . . . . . . . . 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 3843	. . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . 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 1175	. . . . . . . . . . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . . . . . . . . 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 24694	. . . . . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . . . . 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 573	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . . 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 573	. . . . . . . . . . . . . . 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 1334	. . . . . . . . . . . . . 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 398	. . . . . . . . . . . . 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 6501	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑣 → (𝐺‘𝑦) = (𝐺‘𝑣))
196	195	oveq1d 6993	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → ((𝐺‘𝑦) − (𝐺‘𝑧)) = ((𝐺‘𝑣) − (𝐺‘𝑧)))
197	196, 101	oveq12d 6996	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑣 → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)))
198		eqid 2778	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))
199		ovex 7010	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)) ∈ V
200	197, 198, 199	fvmpt 6597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)))‘𝑣) = (((𝐺‘𝑣) − (𝐺‘𝑧)) / (𝑣 − 𝑧)))
201	200	fvoveq1d 7000	. . . . . . . . . . . . . . . 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 4940	. . . . . . . . . . . . . . 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 333	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 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 251	. . . . . . . . . . . 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 3127	. . . . . . . . . . 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 447	. . . . . . . . . 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 639	. . . . . . . . 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 23087	. . . . . . . . . . . 12 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
210		xmetres2 22677	. . . . . . . . . . . 12 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
211	209, 138, 210	sylancr 578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
212	211	ad3antrrr 717	. . . . . . . . . 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 23098	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ Top
214		resttop 21475	. . . . . . . . . . . . . . . . 17 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ∈ {ℝ, ℂ}) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
215	213, 3, 214	sylancr 578	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
216	19	cnfldtopon 23097	. . . . . . . . . . . . . . . . . . 19 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
217		resttopon 21476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
218	216, 11, 217	sylancr 578	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
219		toponuni 21229	. . . . . . . . . . . . . . . . . 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 3899	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
222		eqid 2778	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
223	222	ntrss2 21372	. . . . . . . . . . . . . . . 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 576	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ⊆ 𝑋)
225	224, 26	eqssd 3877	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) = 𝑋)
226	222	isopn3 21381	. . . . . . . . . . . . . . 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 576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ↔ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) = 𝑋))
228	225, 227	mpbird 249	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
229		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
230	19	cnfldtopn 23096	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
231		eqid 2778	. . . . . . . . . . . . . . 15 ⊢ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
232	229, 230, 231	metrest 22840	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
233	209, 11, 232	sylancr 578	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
234	228, 233	eleqtrd 2868	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
235	234	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
236	235	ad3antrrr 717	. . . . . . . . . 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 713	. . . . . . . . . 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 758	. . . . . . . . . 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 22810	. . . . . . . . . 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 1353	. . . . . . . . 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 3220	. . . . . . . 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 3236	. . . . . . 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 3230	. . . . . 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 686	. . . 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 3132	. . 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 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝐺:𝑋⟶ℂ)
248		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
249	247, 139, 248	dvlem 24200	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧)) ∈ ℂ)
250	249	fmpttd 6704	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺‘𝑦) − (𝐺‘𝑧)) / (𝑦 − 𝑧))):(𝑋 ∖ {𝑧})⟶ℂ)
251	139	ssdifssd 4011	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
252	139, 248	sseldd 3861	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ ℂ)
253	250, 251, 252	ellimc3 24183	. . 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 700	. 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 24202	. 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 700	1 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧(𝑆 D 𝐺)(𝐻‘𝑧))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 Vcvv 3415 ∖ cdif 3828 ⊆ wss 3831 {csn 4442 {cpr 4444 ∪ cuni 4713 class class class wbr 4930 ↦ cmpt 5009 × cxp 5406 dom cdm 5408 ↾ cres 5410 ∘ ccom 5412 Rel wrel 5413 Fun wfun 6184 Fn wfn 6185 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ↑_𝑚 cmap 8208 ℂcc 10335 ℝcr 10336 < clt 10476 − cmin 10672 / cdiv 11100 2c2 11498 ℤcz 11796 ℤ_≥cuz 12061 ℝ⁺crp 12207 abscabs 14457 ⇝ cli 14705 ↾_t crest 16553 TopOpenctopn 16554 ∞Metcxmet 20235 ballcbl 20237 MetOpencmopn 20240 ℂ_fldccnfld 20250 Topctop 21208 TopOnctopon 21225 intcnt 21332 lim_ℂ climc 24166 D cdv 24167 ⇝_𝑢culm 24670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-limc 24170 df-dv 24171 df-ulm 24671

This theorem is referenced by: ulmdv 24697

W3C validator