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Theorem ulmdvlem3 24060
Description: Lemma for ulmdv 24061. (Contributed by Mario Carneiro, 8-May-2015.) (Proof shortened by Mario Carneiro, 28-Dec-2016.)
Hypotheses
Ref Expression
ulmdv.z 𝑍 = (ℤ𝑀)
ulmdv.s (𝜑𝑆 ∈ {ℝ, ℂ})
ulmdv.m (𝜑𝑀 ∈ ℤ)
ulmdv.f (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))
ulmdv.g (𝜑𝐺:𝑋⟶ℂ)
ulmdv.l ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))
ulmdv.u (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)
Assertion
Ref Expression
ulmdvlem3 ((𝜑𝑧𝑋) → 𝑧(𝑆 D 𝐺)(𝐻𝑧))
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.
StepHypRef Expression
1 ulmdv.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
2 uzid 11646 . . . . . 6 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
31, 2syl 17 . . . . 5 (𝜑𝑀 ∈ (ℤ𝑀))
4 ulmdv.z . . . . 5 𝑍 = (ℤ𝑀)
53, 4syl6eleqr 2709 . . . 4 (𝜑𝑀𝑍)
6 ulmdv.s . . . . . . 7 (𝜑𝑆 ∈ {ℝ, ℂ})
7 ulmdv.f . . . . . . 7 (𝜑𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))
8 ulmdv.g . . . . . . 7 (𝜑𝐺:𝑋⟶ℂ)
9 ulmdv.l . . . . . . 7 ((𝜑𝑧𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))
10 ulmdv.u . . . . . . 7 (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)
114, 6, 1, 7, 8, 9, 10ulmdvlem2 24059 . . . . . 6 ((𝜑𝑘𝑍) → dom (𝑆 D (𝐹𝑘)) = 𝑋)
12 recnprss 23574 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
136, 12syl 17 . . . . . . . 8 (𝜑𝑆 ⊆ ℂ)
1413adantr 481 . . . . . . 7 ((𝜑𝑘𝑍) → 𝑆 ⊆ ℂ)
157ffvelrnda 6315 . . . . . . . 8 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ (ℂ ↑𝑚 𝑋))
16 elmapi 7823 . . . . . . . 8 ((𝐹𝑘) ∈ (ℂ ↑𝑚 𝑋) → (𝐹𝑘):𝑋⟶ℂ)
1715, 16syl 17 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘):𝑋⟶ℂ)
18 dvbsss 23572 . . . . . . . 8 dom (𝑆 D (𝐹𝑘)) ⊆ 𝑆
1911, 18syl6eqssr 3635 . . . . . . 7 ((𝜑𝑘𝑍) → 𝑋𝑆)
20 eqid 2621 . . . . . . 7 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
21 eqid 2621 . . . . . . 7 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
2214, 17, 19, 20, 21dvbssntr 23570 . . . . . 6 ((𝜑𝑘𝑍) → dom (𝑆 D (𝐹𝑘)) ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))
2311, 22eqsstr3d 3619 . . . . 5 ((𝜑𝑘𝑍) → 𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))
2423ralrimiva 2960 . . . 4 (𝜑 → ∀𝑘𝑍 𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))
25 biidd 252 . . . . 5 (𝑘 = 𝑀 → (𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ↔ 𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)))
2625rspcv 3291 . . . 4 (𝑀𝑍 → (∀𝑘𝑍 𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) → 𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋)))
275, 24, 26sylc 65 . . 3 (𝜑𝑋 ⊆ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))
2827sselda 3583 . 2 ((𝜑𝑧𝑋) → 𝑧 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋))
29 ulmcl 24039 . . . . 5 ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻𝐻:𝑋⟶ℂ)
3010, 29syl 17 . . . 4 (𝜑𝐻:𝑋⟶ℂ)
3130ffvelrnda 6315 . . 3 ((𝜑𝑧𝑋) → (𝐻𝑧) ∈ ℂ)
32 rphalfcl 11802 . . . . . . . 8 (𝑟 ∈ ℝ+ → (𝑟 / 2) ∈ ℝ+)
3332adantl 482 . . . . . . 7 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑟 / 2) ∈ ℝ+)
34 rphalfcl 11802 . . . . . . 7 ((𝑟 / 2) ∈ ℝ+ → ((𝑟 / 2) / 2) ∈ ℝ+)
3533, 34syl 17 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ((𝑟 / 2) / 2) ∈ ℝ+)
36 ulmrel 24036 . . . . . . . . . 10 Rel (⇝𝑢𝑋)
37 releldm 5318 . . . . . . . . . 10 ((Rel (⇝𝑢𝑋) ∧ (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻) → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) ∈ dom (⇝𝑢𝑋))
3836, 10, 37sylancr 694 . . . . . . . . 9 (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) ∈ dom (⇝𝑢𝑋))
39 ulmscl 24037 . . . . . . . . . . 11 ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻𝑋 ∈ V)
4010, 39syl 17 . . . . . . . . . 10 (𝜑𝑋 ∈ V)
41 ovex 6632 . . . . . . . . . . . . 13 (𝑆 D (𝐹𝑘)) ∈ V
4241rgenw 2919 . . . . . . . . . . . 12 𝑘𝑍 (𝑆 D (𝐹𝑘)) ∈ V
43 eqid 2621 . . . . . . . . . . . . 13 (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) = (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))
4443fnmpt 5977 . . . . . . . . . . . 12 (∀𝑘𝑍 (𝑆 D (𝐹𝑘)) ∈ V → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) Fn 𝑍)
4542, 44mp1i 13 . . . . . . . . . . 11 (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) Fn 𝑍)
46 ulmf2 24042 . . . . . . . . . . 11 (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) Fn 𝑍 ∧ (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻) → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))):𝑍⟶(ℂ ↑𝑚 𝑋))
4745, 10, 46syl2anc 692 . . . . . . . . . 10 (𝜑 → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))):𝑍⟶(ℂ ↑𝑚 𝑋))
484, 1, 40, 47ulmcau2 24054 . . . . . . . . 9 (𝜑 → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))) ∈ dom (⇝𝑢𝑋) ↔ ∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠))
4938, 48mpbid 222 . . . . . . . 8 (𝜑 → ∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠)
504uztrn2 11649 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑍𝑛 ∈ (ℤ𝑗)) → 𝑛𝑍)
5150ad2ant2lr 783 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → 𝑛𝑍)
52 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑛 → (𝐹𝑘) = (𝐹𝑛))
5352oveq2d 6620 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑛 → (𝑆 D (𝐹𝑘)) = (𝑆 D (𝐹𝑛)))
54 ovex 6632 . . . . . . . . . . . . . . . . . 18 (𝑆 D (𝐹𝑛)) ∈ V
5553, 43, 54fvmpt 6239 . . . . . . . . . . . . . . . . 17 (𝑛𝑍 → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛) = (𝑆 D (𝐹𝑛)))
5651, 55syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛) = (𝑆 D (𝐹𝑛)))
5756fveq1d 6150 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) = ((𝑆 D (𝐹𝑛))‘𝑥))
58 simprr 795 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → 𝑚 ∈ (ℤ𝑛))
594uztrn2 11649 . . . . . . . . . . . . . . . . . 18 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
6051, 58, 59syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → 𝑚𝑍)
61 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑚 → (𝐹𝑘) = (𝐹𝑚))
6261oveq2d 6620 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑚 → (𝑆 D (𝐹𝑘)) = (𝑆 D (𝐹𝑚)))
63 ovex 6632 . . . . . . . . . . . . . . . . . 18 (𝑆 D (𝐹𝑚)) ∈ V
6462, 43, 63fvmpt 6239 . . . . . . . . . . . . . . . . 17 (𝑚𝑍 → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚) = (𝑆 D (𝐹𝑚)))
6560, 64syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚) = (𝑆 D (𝐹𝑚)))
6665fveq1d 6150 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥) = ((𝑆 D (𝐹𝑚))‘𝑥))
6757, 66oveq12d 6622 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → ((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥)) = (((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥)))
6867fveq2d 6152 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) = (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))))
6968breq1d 4623 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → ((abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠))
7069ralbidv 2980 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ (𝑛 ∈ (ℤ𝑗) ∧ 𝑚 ∈ (ℤ𝑛))) → (∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠))
71702ralbidva 2982 . . . . . . . . . 10 ((𝜑𝑗𝑍) → (∀𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠))
7271rexbidva 3042 . . . . . . . . 9 (𝜑 → (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠))
7372ralbidv 2980 . . . . . . . 8 (𝜑 → (∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘((((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑥) − (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑚)‘𝑥))) < 𝑠 ↔ ∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠))
7449, 73mpbid 222 . . . . . . 7 (𝜑 → ∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠)
7574ad2antrr 761 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠)
76 breq2 4617 . . . . . . . . 9 (𝑠 = ((𝑟 / 2) / 2) → ((abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠 ↔ (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
77762ralbidv 2983 . . . . . . . 8 (𝑠 = ((𝑟 / 2) / 2) → (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠 ↔ ∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
7877rexralbidv 3051 . . . . . . 7 (𝑠 = ((𝑟 / 2) / 2) → (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠 ↔ ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
7978rspcv 3291 . . . . . 6 (((𝑟 / 2) / 2) ∈ ℝ+ → (∀𝑠 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < 𝑠 → ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2)))
8035, 75, 79sylc 65 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2))
811ad2antrr 761 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → 𝑀 ∈ ℤ)
8253fveq1d 6150 . . . . . . . 8 (𝑘 = 𝑛 → ((𝑆 D (𝐹𝑘))‘𝑧) = ((𝑆 D (𝐹𝑛))‘𝑧))
83 eqid 2621 . . . . . . . 8 (𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧)) = (𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧))
84 fvex 6158 . . . . . . . 8 ((𝑆 D (𝐹𝑛))‘𝑧) ∈ V
8582, 83, 84fvmpt 6239 . . . . . . 7 (𝑛𝑍 → ((𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹𝑛))‘𝑧))
8685adantl 482 . . . . . 6 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ((𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧))‘𝑛) = ((𝑆 D (𝐹𝑛))‘𝑧))
8747ad2antrr 761 . . . . . . 7 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘))):𝑍⟶(ℂ ↑𝑚 𝑋))
88 simplr 791 . . . . . . 7 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → 𝑧𝑋)
89 fvex 6158 . . . . . . . . . 10 (ℤ𝑀) ∈ V
904, 89eqeltri 2694 . . . . . . . . 9 𝑍 ∈ V
9190mptex 6440 . . . . . . . 8 (𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧)) ∈ V
9291a1i 11 . . . . . . 7 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧)) ∈ V)
9355adantl 482 . . . . . . . . 9 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛) = (𝑆 D (𝐹𝑛)))
9493fveq1d 6150 . . . . . . . 8 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑧) = ((𝑆 D (𝐹𝑛))‘𝑧))
9594, 86eqtr4d 2658 . . . . . . 7 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛)‘𝑧) = ((𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧))‘𝑛))
9610ad2antrr 761 . . . . . . 7 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))(⇝𝑢𝑋)𝐻)
974, 81, 87, 88, 92, 95, 96ulmclm 24045 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑘𝑍 ↦ ((𝑆 D (𝐹𝑘))‘𝑧)) ⇝ (𝐻𝑧))
984, 81, 33, 86, 97climi2 14176 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2))
994rexanuz2 14023 . . . . . . 7 (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ↔ (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))
1004r19.2uz 14025 . . . . . . 7 (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) → ∃𝑛𝑍 (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))
10199, 100sylbir 225 . . . . . 6 ((∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) → ∃𝑛𝑍 (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))
10235adantr 481 . . . . . . . . . 10 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ((𝑟 / 2) / 2) ∈ ℝ+)
103 simpllr 798 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑧𝑋)
10487ffvelrnda 6315 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ((𝑘𝑍 ↦ (𝑆 D (𝐹𝑘)))‘𝑛) ∈ (ℂ ↑𝑚 𝑋))
10593, 104eqeltrrd 2699 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑆 D (𝐹𝑛)) ∈ (ℂ ↑𝑚 𝑋))
106 elmapi 7823 . . . . . . . . . . . . . . . . 17 ((𝑆 D (𝐹𝑛)) ∈ (ℂ ↑𝑚 𝑋) → (𝑆 D (𝐹𝑛)):𝑋⟶ℂ)
107 fdm 6008 . . . . . . . . . . . . . . . . 17 ((𝑆 D (𝐹𝑛)):𝑋⟶ℂ → dom (𝑆 D (𝐹𝑛)) = 𝑋)
108105, 106, 1073syl 18 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → dom (𝑆 D (𝐹𝑛)) = 𝑋)
109103, 108eleqtrrd 2701 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑧 ∈ dom (𝑆 D (𝐹𝑛)))
1106ad3antrrr 765 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑆 ∈ {ℝ, ℂ})
111 dvfg 23576 . . . . . . . . . . . . . . . 16 (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹𝑛)):dom (𝑆 D (𝐹𝑛))⟶ℂ)
112 ffun 6005 . . . . . . . . . . . . . . . 16 ((𝑆 D (𝐹𝑛)):dom (𝑆 D (𝐹𝑛))⟶ℂ → Fun (𝑆 D (𝐹𝑛)))
113 funfvbrb 6286 . . . . . . . . . . . . . . . 16 (Fun (𝑆 D (𝐹𝑛)) → (𝑧 ∈ dom (𝑆 D (𝐹𝑛)) ↔ 𝑧(𝑆 D (𝐹𝑛))((𝑆 D (𝐹𝑛))‘𝑧)))
114110, 111, 112, 1134syl 19 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑧 ∈ dom (𝑆 D (𝐹𝑛)) ↔ 𝑧(𝑆 D (𝐹𝑛))((𝑆 D (𝐹𝑛))‘𝑧)))
115109, 114mpbid 222 . . . . . . . . . . . . . 14 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑧(𝑆 D (𝐹𝑛))((𝑆 D (𝐹𝑛))‘𝑧))
116 eqid 2621 . . . . . . . . . . . . . . 15 (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))
117110, 12syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑆 ⊆ ℂ)
1187ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → 𝐹:𝑍⟶(ℂ ↑𝑚 𝑋))
119118ffvelrnda 6315 . . . . . . . . . . . . . . . 16 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝐹𝑛) ∈ (ℂ ↑𝑚 𝑋))
120 elmapi 7823 . . . . . . . . . . . . . . . 16 ((𝐹𝑛) ∈ (ℂ ↑𝑚 𝑋) → (𝐹𝑛):𝑋⟶ℂ)
121119, 120syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝐹𝑛):𝑋⟶ℂ)
12219ralrimiva 2960 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝑍 𝑋𝑆)
123 biidd 252 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑀 → (𝑋𝑆𝑋𝑆))
124123rspcv 3291 . . . . . . . . . . . . . . . . 17 (𝑀𝑍 → (∀𝑘𝑍 𝑋𝑆𝑋𝑆))
1255, 122, 124sylc 65 . . . . . . . . . . . . . . . 16 (𝜑𝑋𝑆)
126125ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑋𝑆)
12720, 21, 116, 117, 121, 126eldv 23568 . . . . . . . . . . . . . 14 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑧(𝑆 D (𝐹𝑛))((𝑆 D (𝐹𝑛))‘𝑧) ↔ (𝑧 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))) lim 𝑧))))
128115, 127mpbid 222 . . . . . . . . . . . . 13 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑧 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ ((𝑆 D (𝐹𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))) lim 𝑧)))
129128simprd 479 . . . . . . . . . . . 12 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ((𝑆 D (𝐹𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))) lim 𝑧))
130125adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑋𝑆)
13113adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑧𝑋) → 𝑆 ⊆ ℂ)
132130, 131sstrd 3593 . . . . . . . . . . . . . . . 16 ((𝜑𝑧𝑋) → 𝑋 ⊆ ℂ)
133132ad2antrr 761 . . . . . . . . . . . . . . 15 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑋 ⊆ ℂ)
134121, 133, 103dvlem 23566 . . . . . . . . . . . . . 14 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)) ∈ ℂ)
135134, 116fmptd 6340 . . . . . . . . . . . . 13 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))):(𝑋 ∖ {𝑧})⟶ℂ)
136133ssdifssd 3726 . . . . . . . . . . . . 13 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
137133, 103sseldd 3584 . . . . . . . . . . . . 13 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → 𝑧 ∈ ℂ)
138135, 136, 137ellimc3 23549 . . . . . . . . . . . 12 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (((𝑆 D (𝐹𝑛))‘𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧))) lim 𝑧) ↔ (((𝑆 D (𝐹𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ+𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠))))
139129, 138mpbid 222 . . . . . . . . . . 11 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → (((𝑆 D (𝐹𝑛))‘𝑧) ∈ ℂ ∧ ∀𝑠 ∈ ℝ+𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠)))
140139simprd 479 . . . . . . . . . 10 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ∀𝑠 ∈ ℝ+𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠))
141 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣 → ((𝐹𝑛)‘𝑦) = ((𝐹𝑛)‘𝑣))
142141oveq1d 6619 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) = (((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)))
143 oveq1 6611 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑦𝑧) = (𝑣𝑧))
144142, 143oveq12d 6622 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑣 → ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)) = ((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)))
145 ovex 6632 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) ∈ V
146144, 116, 145fvmpt 6239 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) = ((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)))
147146oveq1d 6619 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧)) = (((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧)))
148147fveq2d 6152 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) = (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))))
149 id 22 . . . . . . . . . . . . . . 15 (𝑠 = ((𝑟 / 2) / 2) → 𝑠 = ((𝑟 / 2) / 2))
150148, 149breqan12rd 4630 . . . . . . . . . . . . . 14 ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠 ↔ (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
151150imbi2d 330 . . . . . . . . . . . . 13 ((𝑠 = ((𝑟 / 2) / 2) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠) ↔ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
152151ralbidva 2979 . . . . . . . . . . . 12 (𝑠 = ((𝑟 / 2) / 2) → (∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠) ↔ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
153152rexbidv 3045 . . . . . . . . . . 11 (𝑠 = ((𝑟 / 2) / 2) → (∃𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠) ↔ ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
154153rspcv 3291 . . . . . . . . . 10 (((𝑟 / 2) / 2) ∈ ℝ+ → (∀𝑠 ∈ ℝ+𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ ((((𝐹𝑛)‘𝑦) − ((𝐹𝑛)‘𝑧)) / (𝑦𝑧)))‘𝑣) − ((𝑆 D (𝐹𝑛))‘𝑧))) < 𝑠) → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2))))
155102, 140, 154sylc 65 . . . . . . . . 9 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ 𝑛𝑍) → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
156155adantrr 752 . . . . . . . 8 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) → ∃𝑤 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
157 anass 680 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ↔ ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+)))
158 df-3an 1038 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))) ↔ ((𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))
159 anass 680 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ↔ (𝜑 ∧ (𝑧𝑋𝑟 ∈ ℝ+)))
1609ralrimiva 2960 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑧𝑋 (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧))
161 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 = 𝑠 → ((𝐹𝑘)‘𝑧) = ((𝐹𝑘)‘𝑠))
162161mpteq2dv 4705 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑠 → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) = (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑠)))
163 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑠 → (𝐺𝑧) = (𝐺𝑠))
164162, 163breq12d 4626 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑠 → ((𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧) ↔ (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑠)) ⇝ (𝐺𝑠)))
165164rspccva 3294 . . . . . . . . . . . . . . . . . . . . . . 23 ((∀𝑧𝑋 (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑧)) ⇝ (𝐺𝑧) ∧ 𝑠𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑠)) ⇝ (𝐺𝑠))
166160, 165sylan 488 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑠𝑋) → (𝑘𝑍 ↦ ((𝐹𝑘)‘𝑠)) ⇝ (𝐺𝑠))
167 simprll 801 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑧𝑋)
168 simprlr 802 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑟 ∈ ℝ+)
169 simprr3 1109 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))))
170 simplll 797 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → 𝑢 ∈ ℝ+)
171169, 170syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑢 ∈ ℝ+)
172 simplr 791 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → 𝑤 ∈ ℝ+)
173169, 172syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑤 ∈ ℝ+)
174 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
175169, 174syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
176175simpld 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑢 < 𝑤)
177175simprd 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)
178 simpr3 1067 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))
179169, 178syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))
180179simprd 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (abs‘(𝑣𝑧)) < 𝑢)
181 simprr1 1107 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑛𝑍)
182 simprr2 1108 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))
183182simpld 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → ∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2))
184182simprd 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2))
185 simpr1 1065 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → 𝑣 ∈ (𝑋 ∖ {𝑧}))
186169, 185syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑣 ∈ (𝑋 ∖ {𝑧}))
187186eldifad 3567 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑣𝑋)
188179simpld 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → 𝑣𝑧)
189 simpr2 1066 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
190169, 189syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
191188, 190mpand 710 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → ((abs‘(𝑣𝑧)) < 𝑤 → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))
1924, 6, 1, 7, 8, 166, 10, 167, 168, 171, 173, 176, 177, 180, 181, 183, 184, 187, 188, 191ulmdvlem1 24058 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ ((𝑧𝑋𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
193192anassrs 679 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑧𝑋𝑟 ∈ ℝ+)) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
194159, 193sylanb 489 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
195158, 194sylan2br 493 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ ((𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
196195anassrs 679 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢)))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
197196anassrs 679 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ ((𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋)) ∧ 𝑤 ∈ ℝ+)) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
198157, 197sylanb 489 . . . . . . . . . . . . . . 15 (((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ (𝑣 ∈ (𝑋 ∖ {𝑧}) ∧ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) ∧ (𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢))) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)
1991983exp2 1282 . . . . . . . . . . . . . 14 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) → (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟))))
200199imp 445 . . . . . . . . . . . . 13 (((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)))
201 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑣 → (𝐺𝑦) = (𝐺𝑣))
202201oveq1d 6619 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑣 → ((𝐺𝑦) − (𝐺𝑧)) = ((𝐺𝑣) − (𝐺𝑧)))
203202, 143oveq12d 6622 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)) = (((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)))
204 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧))) = (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))
205 ovex 6632 . . . . . . . . . . . . . . . . . . 19 (((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) ∈ V
206203, 204, 205fvmpt 6239 . . . . . . . . . . . . . . . . . 18 (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) = (((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)))
207206oveq1d 6619 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧)) = ((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧)))
208207fveq2d 6152 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝑋 ∖ {𝑧}) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) = (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))))
209208breq1d 4623 . . . . . . . . . . . . . . 15 (𝑣 ∈ (𝑋 ∖ {𝑧}) → ((abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟 ↔ (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟))
210209imbi2d 330 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝑋 ∖ {𝑧}) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟) ↔ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)))
211210adantl 482 . . . . . . . . . . . . 13 (((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟) ↔ ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘((((𝐺𝑣) − (𝐺𝑧)) / (𝑣𝑧)) − (𝐻𝑧))) < 𝑟)))
212200, 211sylibrd 249 . . . . . . . . . . . 12 (((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) ∧ 𝑣 ∈ (𝑋 ∖ {𝑧})) → (((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟)))
213212ralimdva 2956 . . . . . . . . . . 11 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ 𝑤 ∈ ℝ+) → (∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟)))
214213impr 648 . . . . . . . . . 10 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
215214an32s 845 . . . . . . . . 9 ((((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) ∧ (𝑢 ∈ ℝ+ ∧ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))) → ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
216 cnxmet 22486 . . . . . . . . . . . 12 (abs ∘ − ) ∈ (∞Met‘ℂ)
217 xmetres2 22076 . . . . . . . . . . . 12 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
218216, 131, 217sylancr 694 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
219218ad3antrrr 765 . . . . . . . . . 10 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆))
22021cnfldtop 22497 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) ∈ Top
221 resttop 20874 . . . . . . . . . . . . . . . . 17 (((TopOpen‘ℂfld) ∈ Top ∧ 𝑆 ∈ {ℝ, ℂ}) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
222220, 6, 221sylancr 694 . . . . . . . . . . . . . . . 16 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top)
22321cnfldtopon 22496 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
224 resttopon 20875 . . . . . . . . . . . . . . . . . . 19 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
225223, 13, 224sylancr 694 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆))
226 toponuni 20642 . . . . . . . . . . . . . . . . . 18 (((TopOpen‘ℂfld) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
227225, 226syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 = ((TopOpen‘ℂfld) ↾t 𝑆))
228125, 227sseqtrd 3620 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ((TopOpen‘ℂfld) ↾t 𝑆))
229 eqid 2621 . . . . . . . . . . . . . . . . 17 ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
230229ntrss2 20771 . . . . . . . . . . . . . . . 16 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝑋 ((TopOpen‘ℂfld) ↾t 𝑆)) → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ⊆ 𝑋)
231222, 228, 230syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ⊆ 𝑋)
232231, 27eqssd 3600 . . . . . . . . . . . . . 14 (𝜑 → ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋)
233229isopn3 20780 . . . . . . . . . . . . . . 15 ((((TopOpen‘ℂfld) ↾t 𝑆) ∈ Top ∧ 𝑋 ((TopOpen‘ℂfld) ↾t 𝑆)) → (𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋))
234222, 228, 233syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆) ↔ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) = 𝑋))
235232, 234mpbird 247 . . . . . . . . . . . . 13 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
236 eqid 2621 . . . . . . . . . . . . . . 15 ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (𝑆 × 𝑆))
23721cnfldtopn 22495 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
238 eqid 2621 . . . . . . . . . . . . . . 15 (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))
239236, 237, 238metrest 22239 . . . . . . . . . . . . . 14 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
240216, 13, 239sylancr 694 . . . . . . . . . . . . 13 (𝜑 → ((TopOpen‘ℂfld) ↾t 𝑆) = (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
241235, 240eleqtrd 2700 . . . . . . . . . . . 12 (𝜑𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
242241adantr 481 . . . . . . . . . . 11 ((𝜑𝑧𝑋) → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
243242ad3antrrr 765 . . . . . . . . . 10 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))))
24488ad2antrr 761 . . . . . . . . . 10 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑧𝑋)
245 simprl 793 . . . . . . . . . 10 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → 𝑤 ∈ ℝ+)
246238mopni3 22209 . . . . . . . . . 10 (((((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (∞Met‘𝑆) ∧ 𝑋 ∈ (MetOpen‘((abs ∘ − ) ↾ (𝑆 × 𝑆))) ∧ 𝑧𝑋) ∧ 𝑤 ∈ ℝ+) → ∃𝑢 ∈ ℝ+ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
247219, 243, 244, 245, 246syl31anc 1326 . . . . . . . . 9 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ+ (𝑢 < 𝑤 ∧ (𝑧(ball‘((abs ∘ − ) ↾ (𝑆 × 𝑆)))𝑢) ⊆ 𝑋))
248215, 247reximddv 3012 . . . . . . . 8 (((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) ∧ (𝑤 ∈ ℝ+ ∧ ∀𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑤) → (abs‘(((((𝐹𝑛)‘𝑣) − ((𝐹𝑛)‘𝑧)) / (𝑣𝑧)) − ((𝑆 D (𝐹𝑛))‘𝑧))) < ((𝑟 / 2) / 2)))) → ∃𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
249156, 248rexlimddv 3028 . . . . . . 7 ((((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) ∧ (𝑛𝑍 ∧ (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)))) → ∃𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
250249rexlimdvaa 3025 . . . . . 6 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → (∃𝑛𝑍 (∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ (abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟)))
251101, 250syl5 34 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ((∃𝑗𝑍𝑛 ∈ (ℤ𝑗)∀𝑚 ∈ (ℤ𝑛)∀𝑥𝑋 (abs‘(((𝑆 D (𝐹𝑛))‘𝑥) − ((𝑆 D (𝐹𝑚))‘𝑥))) < ((𝑟 / 2) / 2) ∧ ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘(((𝑆 D (𝐹𝑛))‘𝑧) − (𝐻𝑧))) < (𝑟 / 2)) → ∃𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟)))
25280, 98, 251mp2and 714 . . . 4 (((𝜑𝑧𝑋) ∧ 𝑟 ∈ ℝ+) → ∃𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
253252ralrimiva 2960 . . 3 ((𝜑𝑧𝑋) → ∀𝑟 ∈ ℝ+𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))
2548adantr 481 . . . . . 6 ((𝜑𝑧𝑋) → 𝐺:𝑋⟶ℂ)
255 simpr 477 . . . . . 6 ((𝜑𝑧𝑋) → 𝑧𝑋)
256254, 132, 255dvlem 23566 . . . . 5 (((𝜑𝑧𝑋) ∧ 𝑦 ∈ (𝑋 ∖ {𝑧})) → (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)) ∈ ℂ)
257256, 204fmptd 6340 . . . 4 ((𝜑𝑧𝑋) → (𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧))):(𝑋 ∖ {𝑧})⟶ℂ)
258132ssdifssd 3726 . . . 4 ((𝜑𝑧𝑋) → (𝑋 ∖ {𝑧}) ⊆ ℂ)
259132, 255sseldd 3584 . . . 4 ((𝜑𝑧𝑋) → 𝑧 ∈ ℂ)
260257, 258, 259ellimc3 23549 . . 3 ((𝜑𝑧𝑋) → ((𝐻𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧))) lim 𝑧) ↔ ((𝐻𝑧) ∈ ℂ ∧ ∀𝑟 ∈ ℝ+𝑢 ∈ ℝ+𝑣 ∈ (𝑋 ∖ {𝑧})((𝑣𝑧 ∧ (abs‘(𝑣𝑧)) < 𝑢) → (abs‘(((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧)))‘𝑣) − (𝐻𝑧))) < 𝑟))))
26131, 253, 260mpbir2and 956 . 2 ((𝜑𝑧𝑋) → (𝐻𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧))) lim 𝑧))
26220, 21, 204, 131, 254, 130eldv 23568 . 2 ((𝜑𝑧𝑋) → (𝑧(𝑆 D 𝐺)(𝐻𝑧) ↔ (𝑧 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑆))‘𝑋) ∧ (𝐻𝑧) ∈ ((𝑦 ∈ (𝑋 ∖ {𝑧}) ↦ (((𝐺𝑦) − (𝐺𝑧)) / (𝑦𝑧))) lim 𝑧))))
26328, 261, 262mpbir2and 956 1 ((𝜑𝑧𝑋) → 𝑧(𝑆 D 𝐺)(𝐻𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  wss 3555  {csn 4148  {cpr 4150   cuni 4402   class class class wbr 4613  cmpt 4673   × cxp 5072  dom cdm 5074  cres 5076  ccom 5078  Rel wrel 5079  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cc 9878  cr 9879   < clt 10018  cmin 10210   / cdiv 10628  2c2 11014  cz 11321  cuz 11631  +crp 11776  abscabs 13908  cli 14149  t crest 16002  TopOpenctopn 16003  ∞Metcxmt 19650  ballcbl 19652  MetOpencmopn 19655  fldccnfld 19665  Topctop 20617  TopOnctopon 20618  intcnt 20731   lim climc 23532   D cdv 23533  𝑢culm 24034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537  df-ulm 24035
This theorem is referenced by:  ulmdv  24061
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