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Theorem lhop1lem 24525
Description: Lemma for lhop1 24526. (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
lhop1.a (𝜑𝐴 ∈ ℝ)
lhop1.b (𝜑𝐵 ∈ ℝ*)
lhop1.l (𝜑𝐴 < 𝐵)
lhop1.f (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)
lhop1.g (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)
lhop1.if (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
lhop1.ig (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
lhop1.f0 (𝜑 → 0 ∈ (𝐹 lim 𝐴))
lhop1.g0 (𝜑 → 0 ∈ (𝐺 lim 𝐴))
lhop1.gn0 (𝜑 → ¬ 0 ∈ ran 𝐺)
lhop1.gd0 (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
lhop1.c (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴))
lhop1lem.e (𝜑𝐸 ∈ ℝ+)
lhop1lem.d (𝜑𝐷 ∈ ℝ)
lhop1lem.db (𝜑𝐷𝐵)
lhop1lem.x (𝜑𝑋 ∈ (𝐴(,)𝐷))
lhop1lem.t (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)
lhop1lem.r 𝑅 = (𝐴 + (𝑟 / 2))
Assertion
Ref Expression
lhop1lem (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) < (2 · 𝐸))
Distinct variable groups:   𝑧,𝑟,𝐵   𝑡,𝐷   𝜑,𝑟,𝑧   𝑧,𝑅   𝑡,𝑟,𝐴,𝑧   𝐸,𝑟,𝑡   𝑋,𝑟,𝑧   𝐶,𝑟,𝑡,𝑧   𝐹,𝑟,𝑡,𝑧   𝐺,𝑟,𝑡,𝑧
Allowed substitution hints:   𝜑(𝑡)   𝐵(𝑡)   𝐷(𝑧,𝑟)   𝑅(𝑡,𝑟)   𝐸(𝑧)   𝑋(𝑡)

Proof of Theorem lhop1lem
Dummy variables 𝑣 𝑥 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lhop1.f . . . . . . 7 (𝜑𝐹:(𝐴(,)𝐵)⟶ℝ)
2 lhop1.b . . . . . . . . 9 (𝜑𝐵 ∈ ℝ*)
3 lhop1lem.db . . . . . . . . 9 (𝜑𝐷𝐵)
4 iooss2 12767 . . . . . . . . 9 ((𝐵 ∈ ℝ*𝐷𝐵) → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
52, 3, 4syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
6 lhop1lem.x . . . . . . . 8 (𝜑𝑋 ∈ (𝐴(,)𝐷))
75, 6sseldd 3971 . . . . . . 7 (𝜑𝑋 ∈ (𝐴(,)𝐵))
81, 7ffvelrnd 6847 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ ℝ)
98recnd 10661 . . . . 5 (𝜑 → (𝐹𝑋) ∈ ℂ)
10 lhop1.g . . . . . . 7 (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)
1110, 7ffvelrnd 6847 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ ℝ)
1211recnd 10661 . . . . 5 (𝜑 → (𝐺𝑋) ∈ ℂ)
13 lhop1.gn0 . . . . . 6 (𝜑 → ¬ 0 ∈ ran 𝐺)
1410ffnd 6511 . . . . . . . . 9 (𝜑𝐺 Fn (𝐴(,)𝐵))
15 fnfvelrn 6843 . . . . . . . . 9 ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑋 ∈ (𝐴(,)𝐵)) → (𝐺𝑋) ∈ ran 𝐺)
1614, 7, 15syl2anc 584 . . . . . . . 8 (𝜑 → (𝐺𝑋) ∈ ran 𝐺)
17 eleq1 2904 . . . . . . . 8 ((𝐺𝑋) = 0 → ((𝐺𝑋) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
1816, 17syl5ibcom 246 . . . . . . 7 (𝜑 → ((𝐺𝑋) = 0 → 0 ∈ ran 𝐺))
1918necon3bd 3034 . . . . . 6 (𝜑 → (¬ 0 ∈ ran 𝐺 → (𝐺𝑋) ≠ 0))
2013, 19mpd 15 . . . . 5 (𝜑 → (𝐺𝑋) ≠ 0)
219, 12, 20divcld 11408 . . . 4 (𝜑 → ((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ)
22 limccl 24388 . . . . 5 ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴) ⊆ ℂ
23 lhop1.c . . . . 5 (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴))
2422, 23sseldi 3968 . . . 4 (𝜑𝐶 ∈ ℂ)
2521, 24subcld 10989 . . 3 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) − 𝐶) ∈ ℂ)
2625abscld 14789 . 2 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ∈ ℝ)
27 lhop1lem.e . . 3 (𝜑𝐸 ∈ ℝ+)
2827rpred 12424 . 2 (𝜑𝐸 ∈ ℝ)
29 2re 11703 . . . 4 2 ∈ ℝ
3029a1i 11 . . 3 (𝜑 → 2 ∈ ℝ)
3130, 28remulcld 10663 . 2 (𝜑 → (2 · 𝐸) ∈ ℝ)
32 cnxmet 23296 . . . . . . . . . . . . 13 (abs ∘ − ) ∈ (∞Met‘ℂ)
3332a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
34 simprl 767 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → 𝑣 ∈ (TopOpen‘ℂfld))
35 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → 𝐴𝑣)
36 eliooord 12789 . . . . . . . . . . . . . . . 16 (𝑋 ∈ (𝐴(,)𝐷) → (𝐴 < 𝑋𝑋 < 𝐷))
376, 36syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 < 𝑋𝑋 < 𝐷))
3837simpld 495 . . . . . . . . . . . . . 14 (𝜑𝐴 < 𝑋)
39 lhop1.a . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
40 ioossre 12791 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐷) ⊆ ℝ
4140, 6sseldi 3968 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ ℝ)
42 difrp 12420 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (𝐴 < 𝑋 ↔ (𝑋𝐴) ∈ ℝ+))
4339, 41, 42syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 < 𝑋 ↔ (𝑋𝐴) ∈ ℝ+))
4438, 43mpbid 233 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐴) ∈ ℝ+)
4544adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → (𝑋𝐴) ∈ ℝ+)
46 eqid 2825 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4746cnfldtopn 23305 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
4847mopni3 23019 . . . . . . . . . . . 12 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣) ∧ (𝑋𝐴) ∈ ℝ+) → ∃𝑟 ∈ ℝ+ (𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣))
4933, 34, 35, 45, 48syl31anc 1367 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑟 ∈ ℝ+ (𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣))
50 ssrin 4213 . . . . . . . . . . . . . . . 16 ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ⊆ (𝑣 ∩ (𝐴(,)𝑋)))
51 lbioo 12762 . . . . . . . . . . . . . . . . . . 19 ¬ 𝐴 ∈ (𝐴(,)𝑋)
52 disjsn 4645 . . . . . . . . . . . . . . . . . . 19 (((𝐴(,)𝑋) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ (𝐴(,)𝑋))
5351, 52mpbir 232 . . . . . . . . . . . . . . . . . 18 ((𝐴(,)𝑋) ∩ {𝐴}) = ∅
54 disj3 4405 . . . . . . . . . . . . . . . . . 18 (((𝐴(,)𝑋) ∩ {𝐴}) = ∅ ↔ (𝐴(,)𝑋) = ((𝐴(,)𝑋) ∖ {𝐴}))
5553, 54mpbi 231 . . . . . . . . . . . . . . . . 17 (𝐴(,)𝑋) = ((𝐴(,)𝑋) ∖ {𝐴})
5655ineq2i 4189 . . . . . . . . . . . . . . . 16 (𝑣 ∩ (𝐴(,)𝑋)) = (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))
5750, 56syl6sseq 4020 . . . . . . . . . . . . . . 15 ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ⊆ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))
58 lhop1lem.r . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 = (𝐴 + (𝑟 / 2))
5939adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℝ)
60 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ+)
6160rpred 12424 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ)
6261rehalfcld 11876 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ)
6359, 62readdcld 10662 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴 + (𝑟 / 2)) ∈ ℝ)
6458, 63eqeltrid 2921 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℝ)
6564recnd 10661 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℂ)
6639recnd 10661 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ ℂ)
6766adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℂ)
68 eqid 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (abs ∘ − ) = (abs ∘ − )
6968cnmetdval 23294 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑅(abs ∘ − )𝐴) = (abs‘(𝑅𝐴)))
7065, 67, 69syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) = (abs‘(𝑅𝐴)))
7158oveq1i 7161 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅𝐴) = ((𝐴 + (𝑟 / 2)) − 𝐴)
7261recnd 10661 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℂ)
7372halfcld 11874 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℂ)
7467, 73pncan2d 10991 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐴 + (𝑟 / 2)) − 𝐴) = (𝑟 / 2))
7571, 74syl5eq 2872 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅𝐴) = (𝑟 / 2))
7675fveq2d 6670 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘(𝑅𝐴)) = (abs‘(𝑟 / 2)))
7760rphalfcld 12436 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ+)
7877rpred 12424 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ)
7977rpge0d 12428 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 0 ≤ (𝑟 / 2))
8078, 79absidd 14775 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘(𝑟 / 2)) = (𝑟 / 2))
8170, 76, 803eqtrd 2864 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) = (𝑟 / 2))
82 rphalflt 12411 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ+ → (𝑟 / 2) < 𝑟)
8360, 82syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) < 𝑟)
8481, 83eqbrtrd 5084 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) < 𝑟)
8532a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs ∘ − ) ∈ (∞Met‘ℂ))
8661rexrd 10683 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ*)
87 elbl3 22917 . . . . . . . . . . . . . . . . . . . 20 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ)) → (𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐴) < 𝑟))
8885, 86, 67, 65, 87syl22anc 836 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐴) < 𝑟))
8984, 88mpbird 258 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟))
9059, 77ltaddrpd 12457 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 < (𝐴 + (𝑟 / 2)))
9190, 58breqtrrdi 5104 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 < 𝑅)
9241adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 ∈ ℝ)
9392, 59resubcld 11060 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑋𝐴) ∈ ℝ)
94 simprr 769 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 < (𝑋𝐴))
9578, 61, 93, 83, 94lttrd 10793 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) < (𝑋𝐴))
9659, 78, 92ltaddsub2d 11233 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐴 + (𝑟 / 2)) < 𝑋 ↔ (𝑟 / 2) < (𝑋𝐴)))
9795, 96mpbird 258 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴 + (𝑟 / 2)) < 𝑋)
9858, 97eqbrtrid 5097 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 < 𝑋)
9959rexrd 10683 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℝ*)
10041rexrd 10683 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑋 ∈ ℝ*)
101100adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 ∈ ℝ*)
102 elioo2 12772 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℝ*𝑋 ∈ ℝ*) → (𝑅 ∈ (𝐴(,)𝑋) ↔ (𝑅 ∈ ℝ ∧ 𝐴 < 𝑅𝑅 < 𝑋)))
10399, 101, 102syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅 ∈ (𝐴(,)𝑋) ↔ (𝑅 ∈ ℝ ∧ 𝐴 < 𝑅𝑅 < 𝑋)))
10464, 91, 98, 103mpbir3and 1336 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(,)𝑋))
10589, 104elind 4174 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)))
1069adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑋) ∈ ℂ)
1071adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
108 lhop1lem.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐷 ∈ ℝ)
109108rexrd 10683 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐷 ∈ ℝ*)
11037simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑋 < 𝐷)
11141, 108, 110ltled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑋𝐷)
112100, 109, 2, 111, 3xrletrd 12548 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑋𝐵)
113 iooss2 12767 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ ℝ*𝑋𝐵) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
1142, 112, 113syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
115114adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
116115, 104sseldd 3971 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(,)𝐵))
117107, 116ffvelrnd 6847 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑅) ∈ ℝ)
118117recnd 10661 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑅) ∈ ℂ)
119106, 118subcld 10989 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐹𝑋) − (𝐹𝑅)) ∈ ℂ)
12012adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑋) ∈ ℂ)
12110adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
122121, 116ffvelrnd 6847 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑅) ∈ ℝ)
123122recnd 10661 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑅) ∈ ℂ)
124120, 123subcld 10989 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐺𝑋) − (𝐺𝑅)) ∈ ℂ)
125 fveq2 6666 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑅 → (𝐺𝑧) = (𝐺𝑅))
126125oveq2d 7167 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑅 → ((𝐺𝑋) − (𝐺𝑧)) = ((𝐺𝑋) − (𝐺𝑅)))
127126neeq1d 3079 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝑅 → (((𝐺𝑋) − (𝐺𝑧)) ≠ 0 ↔ ((𝐺𝑋) − (𝐺𝑅)) ≠ 0))
128 lhop1.gd0 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
129128adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐺))
13012adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑋) ∈ ℂ)
131114sselda 3970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ (𝐴(,)𝐵))
13210ffvelrnda 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑧 ∈ (𝐴(,)𝐵)) → (𝐺𝑧) ∈ ℝ)
133131, 132syldan 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑧) ∈ ℝ)
134133recnd 10661 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑧) ∈ ℂ)
135130, 134subeq0ad 10999 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑧)) = 0 ↔ (𝐺𝑋) = (𝐺𝑧)))
136 ioossre 12791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐴(,)𝐵) ⊆ ℝ
137136, 131sseldi 3968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ ℝ)
138137adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑧 ∈ ℝ)
13941ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑋 ∈ ℝ)
140 eliooord 12789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝐴(,)𝑋) → (𝐴 < 𝑧𝑧 < 𝑋))
141140adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐴 < 𝑧𝑧 < 𝑋))
142141simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 < 𝑋)
143142adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑧 < 𝑋)
14439rexrd 10683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝐴 ∈ ℝ*)
145144adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐴 ∈ ℝ*)
1462adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐵 ∈ ℝ*)
147141simpld 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐴 < 𝑧)
148100, 109, 2, 110, 3xrltletrd 12547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝑋 < 𝐵)
149148adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 < 𝐵)
150 iccssioo 12798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 < 𝑧𝑋 < 𝐵)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
151145, 146, 147, 149, 150syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
152151adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
153 ax-resscn 10586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ℝ ⊆ ℂ
154153a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → ℝ ⊆ ℂ)
155 fss 6523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
15610, 153, 155sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑𝐺:(𝐴(,)𝐵)⟶ℂ)
157136a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
158 lhop1.ig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
159 dvcn 24433 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((ℝ ⊆ ℂ ∧ 𝐺:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
160154, 156, 157, 158, 159syl31anc 1367 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
161 cncffvrn 23421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((ℝ ⊆ ℂ ∧ 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐺:(𝐴(,)𝐵)⟶ℝ))
162153, 160, 161sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐺:(𝐴(,)𝐵)⟶ℝ))
16310, 162mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
164163ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
165 rescncf 23420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑧[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐺 ↾ (𝑧[,]𝑋)) ∈ ((𝑧[,]𝑋)–cn→ℝ)))
166152, 164, 165sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝐺 ↾ (𝑧[,]𝑋)) ∈ ((𝑧[,]𝑋)–cn→ℝ))
167153a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ℝ ⊆ ℂ)
168156ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
169136a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝐴(,)𝐵) ⊆ ℝ)
170152, 136syl6ss 3982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧[,]𝑋) ⊆ ℝ)
17146tgioo2 23326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
17246, 171dvres 24424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((ℝ ⊆ ℂ ∧ 𝐺:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝑧[,]𝑋) ⊆ ℝ)) → (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋))))
173167, 168, 169, 170, 172syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋))))
174 iccntr 23344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋)) = (𝑧(,)𝑋))
175138, 139, 174syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋)) = (𝑧(,)𝑋))
176175reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
177173, 176eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
178177dmeqd 5772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
179 ioossicc 12815 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧(,)𝑋) ⊆ (𝑧[,]𝑋)
180179, 152sstrid 3981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧(,)𝑋) ⊆ (𝐴(,)𝐵))
181158ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
182180, 181sseqtrrd 4011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧(,)𝑋) ⊆ dom (ℝ D 𝐺))
183 ssdmres 5874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑧(,)𝑋) ⊆ dom (ℝ D 𝐺) ↔ dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)) = (𝑧(,)𝑋))
184182, 183sylib 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)) = (𝑧(,)𝑋))
185178, 184eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = (𝑧(,)𝑋))
186137rexrd 10683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ ℝ*)
187100adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ ℝ*)
18841adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ ℝ)
189137, 188, 142ltled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧𝑋)
190 ubicc2 12846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ*𝑋 ∈ ℝ*𝑧𝑋) → 𝑋 ∈ (𝑧[,]𝑋))
191186, 187, 189, 190syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ (𝑧[,]𝑋))
192191fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = (𝐺𝑋))
193 lbicc2 12845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ*𝑋 ∈ ℝ*𝑧𝑋) → 𝑧 ∈ (𝑧[,]𝑋))
194186, 187, 189, 193syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ (𝑧[,]𝑋))
195194fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) = (𝐺𝑧))
196192, 195eqeq12d 2841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) ↔ (𝐺𝑋) = (𝐺𝑧)))
197196biimpar 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧))
198197eqcomd 2831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋))
199138, 139, 143, 166, 185, 198rolle 24502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ∃𝑤 ∈ (𝑧(,)𝑋)((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0)
200177fveq1d 6668 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = (((ℝ D 𝐺) ↾ (𝑧(,)𝑋))‘𝑤))
201 fvres 6685 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑤 ∈ (𝑧(,)𝑋) → (((ℝ D 𝐺) ↾ (𝑧(,)𝑋))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
202200, 201sylan9eq 2880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
203 dvf 24420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
204158feq2d 6496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝜑 → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
205203, 204mpbii 234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑 → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
206205ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
207206ffnd 6511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
208207adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
209180sselda 3970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐵))
210 fnfvelrn 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
211208, 209, 210syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
212202, 211eqeltrd 2917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) ∈ ran (ℝ D 𝐺))
213 eleq1 2904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
214212, 213syl5ibcom 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
215214rexlimdva 3288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (∃𝑤 ∈ (𝑧(,)𝑋)((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
216199, 215mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 0 ∈ ran (ℝ D 𝐺))
217216ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) = (𝐺𝑧) → 0 ∈ ran (ℝ D 𝐺)))
218135, 217sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑧)) = 0 → 0 ∈ ran (ℝ D 𝐺)))
219218necon3bd 3034 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((𝐺𝑋) − (𝐺𝑧)) ≠ 0))
220129, 219mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
221220ralrimiva 3186 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑧 ∈ (𝐴(,)𝑋)((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
222221adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∀𝑧 ∈ (𝐴(,)𝑋)((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
223127, 222, 104rspcdva 3628 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐺𝑋) − (𝐺𝑅)) ≠ 0)
224119, 124, 223divcld 11408 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) ∈ ℂ)
22524adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐶 ∈ ℂ)
226224, 225subcld 10989 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶) ∈ ℂ)
227226abscld 14789 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ∈ ℝ)
22828adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐸 ∈ ℝ)
229109adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐷 ∈ ℝ*)
230110adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 < 𝐷)
231 iccssioo 12798 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ ℝ*𝐷 ∈ ℝ*) ∧ (𝐴 < 𝑅𝑋 < 𝐷)) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐷))
23299, 229, 91, 230, 231syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐷))
2335adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
234232, 233sstrd 3980 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐵))
235 fss 6523 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
2361, 153, 235sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:(𝐴(,)𝐵)⟶ℂ)
237 lhop1.if . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
238 dvcn 24433 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
239154, 236, 157, 237, 238syl31anc 1367 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
240 cncffvrn 23421 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
241153, 239, 240sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
2421, 241mpbird 258 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
243242adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
244 rescncf 23420 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐹 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ)))
245234, 243, 244sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ))
246163adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
247 rescncf 23420 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐺 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ)))
248234, 246, 247sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ))
249153a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ℝ ⊆ ℂ)
250236adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
251136a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝐵) ⊆ ℝ)
252 iccssre 12811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (𝑅[,]𝑋) ⊆ ℝ)
25364, 92, 252syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ ℝ)
25446, 171dvres 24424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝑅[,]𝑋) ⊆ ℝ)) → (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
255249, 250, 251, 253, 254syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
256 iccntr 23344 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋)) = (𝑅(,)𝑋))
25764, 92, 256syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋)) = (𝑅(,)𝑋))
258257reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
259255, 258eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
260259dmeqd 5772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
26159, 64, 91ltled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴𝑅)
262 iooss1 12766 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℝ*𝐴𝑅) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝑋))
26399, 261, 262syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝑋))
264111adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋𝐷)
265 iooss2 12767 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐷 ∈ ℝ*𝑋𝐷) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐷))
266229, 264, 265syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐷))
267263, 266sstrd 3980 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝐷))
268267, 233sstrd 3980 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝐵))
269237adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
270268, 269sseqtrrd 4011 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ dom (ℝ D 𝐹))
271 ssdmres 5874 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅(,)𝑋) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
272270, 271sylib 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
273260, 272eqtrd 2860 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = (𝑅(,)𝑋))
274156adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
27546, 171dvres 24424 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((ℝ ⊆ ℂ ∧ 𝐺:(𝐴(,)𝐵)⟶ℂ) ∧ ((𝐴(,)𝐵) ⊆ ℝ ∧ (𝑅[,]𝑋) ⊆ ℝ)) → (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
276249, 274, 251, 253, 275syl22anc 836 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
277257reseq2d 5851 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
278276, 277eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
279278dmeqd 5772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
280158adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
281268, 280sseqtrrd 4011 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ dom (ℝ D 𝐺))
282 ssdmres 5874 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅(,)𝑋) ⊆ dom (ℝ D 𝐺) ↔ dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
283281, 282sylib 219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
284279, 283eqtrd 2860 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = (𝑅(,)𝑋))
28564, 92, 98, 245, 248, 273, 284cmvth 24503 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑤 ∈ (𝑅(,)𝑋)((((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤)) = ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)))
28664rexrd 10683 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℝ*)
287286adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅 ∈ ℝ*)
288100ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑋 ∈ ℝ*)
28964, 92, 98ltled 10780 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅𝑋)
290289adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅𝑋)
291 ubicc2 12846 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ ℝ*𝑋 ∈ ℝ*𝑅𝑋) → 𝑋 ∈ (𝑅[,]𝑋))
292287, 288, 290, 291syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑋 ∈ (𝑅[,]𝑋))
293292fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐹𝑋))
294 lbicc2 12845 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ ℝ*𝑋 ∈ ℝ*𝑅𝑋) → 𝑅 ∈ (𝑅[,]𝑋))
295287, 288, 290, 294syl3anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅 ∈ (𝑅[,]𝑋))
296295fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐹𝑅))
297293, 296oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) = ((𝐹𝑋) − (𝐹𝑅)))
298278fveq1d 6668 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤) = (((ℝ D 𝐺) ↾ (𝑅(,)𝑋))‘𝑤))
299 fvres 6685 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ (𝑅(,)𝑋) → (((ℝ D 𝐺) ↾ (𝑅(,)𝑋))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
300298, 299sylan9eq 2880 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
301297, 300oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤)) = (((𝐹𝑋) − (𝐹𝑅)) · ((ℝ D 𝐺)‘𝑤)))
302292fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐺𝑋))
303295fvresd 6686 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐺𝑅))
304302, 303oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) = ((𝐺𝑋) − (𝐺𝑅)))
305259fveq1d 6668 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤) = (((ℝ D 𝐹) ↾ (𝑅(,)𝑋))‘𝑤))
306 fvres 6685 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑅(,)𝑋) → (((ℝ D 𝐹) ↾ (𝑅(,)𝑋))‘𝑤) = ((ℝ D 𝐹)‘𝑤))
307305, 306sylan9eq 2880 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤) = ((ℝ D 𝐹)‘𝑤))
308304, 307oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)) = (((𝐺𝑋) − (𝐺𝑅)) · ((ℝ D 𝐹)‘𝑤)))
309124adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑅)) ∈ ℂ)
310 dvf 24420 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
311237feq2d 6496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
312310, 311mpbii 234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
313312ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
314268sselda 3970 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐵))
315313, 314ffvelrnd 6847 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐹)‘𝑤) ∈ ℂ)
316309, 315mulcomd 10654 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑅)) · ((ℝ D 𝐹)‘𝑤)) = (((ℝ D 𝐹)‘𝑤) · ((𝐺𝑋) − (𝐺𝑅))))
317308, 316eqtrd 2860 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)) = (((ℝ D 𝐹)‘𝑤) · ((𝐺𝑋) − (𝐺𝑅))))
318301, 317eqeq12d 2841 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤)) = ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)) ↔ (((𝐹𝑋) − (𝐹𝑅)) · ((ℝ D 𝐺)‘𝑤)) = (((ℝ D 𝐹)‘𝑤) · ((𝐺𝑋) − (𝐺𝑅)))))
319119adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹𝑋) − (𝐹𝑅)) ∈ ℂ)
320205ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
321320, 314ffvelrnd 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ℂ)
322223adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑅)) ≠ 0)
323128ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐺))
324320ffnd 6511 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
325324, 314, 210syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
326 eleq1 2904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ℝ D 𝐺)‘𝑤) = 0 → (((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
327325, 326syl5ibcom 246 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((ℝ D 𝐺)‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
328327necon3bd 3034 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((ℝ D 𝐺)‘𝑤) ≠ 0))
329323, 328mpd 15 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ≠ 0)
330319, 309, 315, 321, 322, 329divmuleqd 11454 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) ↔ (((𝐹𝑋) − (𝐹𝑅)) · ((ℝ D 𝐺)‘𝑤)) = (((ℝ D 𝐹)‘𝑤) · ((𝐺𝑋) − (𝐺𝑅)))))
331318, 330bitr4d 283 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤)) = ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)) ↔ (((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤))))
332331rexbidva 3300 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (∃𝑤 ∈ (𝑅(,)𝑋)((((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤)) = ((((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) · ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤)) ↔ ∃𝑤 ∈ (𝑅(,)𝑋)(((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤))))
333285, 332mpbid 233 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑤 ∈ (𝑅(,)𝑋)(((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)))
334 fveq2 6666 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑤 → ((ℝ D 𝐹)‘𝑡) = ((ℝ D 𝐹)‘𝑤))
335 fveq2 6666 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑤 → ((ℝ D 𝐺)‘𝑡) = ((ℝ D 𝐺)‘𝑤))
336334, 335oveq12d 7169 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑤 → (((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)))
337336fvoveq1d 7173 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑤 → (abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) = (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)))
338337breq1d 5072 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = 𝑤 → ((abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸 ↔ (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸))
339 lhop1lem.t . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)
340339ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)
341267sselda 3970 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐷))
342338, 340, 341rspcdva 3628 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸)
343 fvoveq1 7174 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) = (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)))
344343breq1d 5072 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) → ((abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) < 𝐸 ↔ (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸))
345342, 344syl5ibrcom 248 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) < 𝐸))
346345rexlimdva 3288 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (∃𝑤 ∈ (𝑅(,)𝑋)(((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) < 𝐸))
347333, 346mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) < 𝐸)
348227, 228, 347ltled 10780 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ≤ 𝐸)
349 fveq2 6666 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑅 → (𝐹𝑢) = (𝐹𝑅))
350349oveq2d 7167 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑅 → ((𝐹𝑋) − (𝐹𝑢)) = ((𝐹𝑋) − (𝐹𝑅)))
351 fveq2 6666 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑅 → (𝐺𝑢) = (𝐺𝑅))
352351oveq2d 7167 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑅 → ((𝐺𝑋) − (𝐺𝑢)) = ((𝐺𝑋) − (𝐺𝑅)))
353350, 352oveq12d 7169 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑅 → (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) = (((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))))
354353fvoveq1d 7173 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑅 → (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) = (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)))
355354breq1d 5072 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑅 → ((abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸 ↔ (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ≤ 𝐸))
356355rspcev 3626 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ∧ (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ≤ 𝐸) → ∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
357105, 348, 356syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
358357adantlr 711 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
359 ssrexv 4037 . . . . . . . . . . . . . . 15 (((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ⊆ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) → (∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸 → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
36057, 358, 359syl2imc 41 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
361360anassrs 468 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) ∧ 𝑟 ∈ ℝ+) ∧ 𝑟 < (𝑋𝐴)) → ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
362361expimpd 454 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) ∧ 𝑟 ∈ ℝ+) → ((𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
363362rexlimdva 3288 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → (∃𝑟 ∈ ℝ+ (𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
36449, 363mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
365 inss2 4209 . . . . . . . . . . . . . 14 (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) ⊆ ((𝐴(,)𝑋) ∖ {𝐴})
366 difss 4111 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ∖ {𝐴}) ⊆ (𝐴(,)𝑋)
367365, 366sstri 3979 . . . . . . . . . . . . 13 (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) ⊆ (𝐴(,)𝑋)
368367sseli 3966 . . . . . . . . . . . 12 (𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) → 𝑢 ∈ (𝐴(,)𝑋))
369 fveq2 6666 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → (𝐹𝑧) = (𝐹𝑢))
370369oveq2d 7167 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑢 → ((𝐹𝑋) − (𝐹𝑧)) = ((𝐹𝑋) − (𝐹𝑢)))
371 fveq2 6666 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → (𝐺𝑧) = (𝐺𝑢))
372371oveq2d 7167 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑢 → ((𝐺𝑋) − (𝐺𝑧)) = ((𝐺𝑋) − (𝐺𝑢)))
373370, 372oveq12d 7169 . . . . . . . . . . . . . . 15 (𝑧 = 𝑢 → (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) = (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))))
374 eqid 2825 . . . . . . . . . . . . . . 15 (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))
375 ovex 7184 . . . . . . . . . . . . . . 15 (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) ∈ V
376373, 374, 375fvmpt 6764 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝑋) → ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) = (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))))
377376fvoveq1d 7173 . . . . . . . . . . . . 13 (𝑢 ∈ (𝐴(,)𝑋) → (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) = (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)))
378377breq1d 5072 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝑋) → ((abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸 ↔ (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
379368, 378syl 17 . . . . . . . . . . 11 (𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) → ((abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸 ↔ (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
380379rexbiia 3250 . . . . . . . . . 10 (∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸 ↔ ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
381364, 380sylibr 235 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸)
382 ovex 7184 . . . . . . . . . . 11 (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) ∈ V
383382, 374fnmpti 6487 . . . . . . . . . 10 (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) Fn (𝐴(,)𝑋)
384 fvoveq1 7174 . . . . . . . . . . . 12 (𝑥 = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) → (abs‘(𝑥𝐶)) = (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)))
385384breq1d 5072 . . . . . . . . . . 11 (𝑥 = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) → ((abs‘(𝑥𝐶)) ≤ 𝐸 ↔ (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸))
386385rexima 6996 . . . . . . . . . 10 (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) Fn (𝐴(,)𝑋) ∧ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) ⊆ (𝐴(,)𝑋)) → (∃𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))(abs‘(𝑥𝐶)) ≤ 𝐸 ↔ ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸))
387383, 367, 386mp2an 688 . . . . . . . . 9 (∃𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))(abs‘(𝑥𝐶)) ≤ 𝐸 ↔ ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸)
388381, 387sylibr 235 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))(abs‘(𝑥𝐶)) ≤ 𝐸)
389 dfrex2 3243 . . . . . . . 8 (∃𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))(abs‘(𝑥𝐶)) ≤ 𝐸 ↔ ¬ ∀𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ¬ (abs‘(𝑥𝐶)) ≤ 𝐸)
390388, 389sylib 219 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ¬ ∀𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ¬ (abs‘(𝑥𝐶)) ≤ 𝐸)
391 ssrab 4052 . . . . . . . 8 (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ↔ (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ ℂ ∧ ∀𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ¬ (abs‘(𝑥𝐶)) ≤ 𝐸))
392391simprbi 497 . . . . . . 7 (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → ∀𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ¬ (abs‘(𝑥𝐶)) ≤ 𝐸)
393390, 392nsyl 142 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})
394393expr 457 . . . . 5 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐴𝑣 → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
395394ralrimiva 3186 . . . 4 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
396 ralinexa 3268 . . . 4 (∀𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}) ↔ ¬ ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
397395, 396sylib 219 . . 3 (𝜑 → ¬ ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
398 fvoveq1 7174 . . . . . . . 8 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → (abs‘(𝑥𝐶)) = (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)))
399398breq1d 5072 . . . . . . 7 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → ((abs‘(𝑥𝐶)) ≤ 𝐸 ↔ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
400399notbid 319 . . . . . 6 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → (¬ (abs‘(𝑥𝐶)) ≤ 𝐸 ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
401400elrab3 3684 . . . . 5 (((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ → (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
40221, 401syl 17 . . . 4 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
403 eleq2 2905 . . . . . 6 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → (((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 ↔ ((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
404 sseq2 3996 . . . . . . . 8 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢 ↔ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
405404anbi2d 628 . . . . . . 7 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → ((𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢) ↔ (𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})))
406405rexbidv 3301 . . . . . 6 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → (∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢) ↔ ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})))
407403, 406imbi12d 346 . . . . 5 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → ((((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢)) ↔ (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))))
4089adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑋) ∈ ℂ)
4091ffvelrnda 6846 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴(,)𝐵)) → (𝐹𝑧) ∈ ℝ)
410131, 409syldan 591 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑧) ∈ ℝ)
411410recnd 10661 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑧) ∈ ℂ)
412408, 411subcld 10989 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐹𝑋) − (𝐹𝑧)) ∈ ℂ)
413130, 134subcld 10989 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ∈ ℂ)
414 eldifsn 4717 . . . . . . . . 9 (((𝐺𝑋) − (𝐺𝑧)) ∈ (ℂ ∖ {0}) ↔ (((𝐺𝑋) − (𝐺𝑧)) ∈ ℂ ∧ ((𝐺𝑋) − (𝐺𝑧)) ≠ 0))
415413, 220, 414sylanbrc 583 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ∈ (ℂ ∖ {0}))
416 ssidd 3993 . . . . . . . 8 (𝜑 → ℂ ⊆ ℂ)
417 difss 4111 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
418417a1i 11 . . . . . . . 8 (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
41946cnfldtopon 23306 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
420 cnex 10610 . . . . . . . . . 10 ℂ ∈ V
421420difexi 5228 . . . . . . . . . 10 (ℂ ∖ {0}) ∈ V
422 txrest 22155 . . . . . . . . . 10 ((((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) ∧ (ℂ ∈ V ∧ (ℂ ∖ {0}) ∈ V)) → (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × (ℂ ∖ {0}))) = (((TopOpen‘ℂfld) ↾t ℂ) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))))
423419, 419, 420, 421, 422mp4an 689 . . . . . . . . 9 (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × (ℂ ∖ {0}))) = (((TopOpen‘ℂfld) ↾t ℂ) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
424 unicntop 23309 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
425424restid 16699 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
426419, 425ax-mp 5 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
427426oveq1i 7161 . . . . . . . . 9 (((TopOpen‘ℂfld) ↾t ℂ) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) = ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
428423, 427eqtr2i 2849 . . . . . . . 8 ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) = (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × (ℂ ∖ {0})))
4299subid1d 10978 . . . . . . . . 9 (𝜑 → ((𝐹𝑋) − 0) = (𝐹𝑋))
430 txtopon 22115 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ∈ (TopOn‘(ℂ × ℂ)))
431419, 419, 430mp2an 688 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ∈ (TopOn‘(ℂ × ℂ))
432431toponrestid 21445 . . . . . . . . . 10 ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × ℂ))
433 limcresi 24398 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴) ⊆ (((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴)
434 ioossre 12791 . . . . . . . . . . . . . 14 (𝐴(,)𝑋) ⊆ ℝ
435 resmpt 5903 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)))
436434, 435ax-mp 5 . . . . . . . . . . . . 13 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋))
437436oveq1i 7161 . . . . . . . . . . . 12 (((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴)
438433, 437sseqtri 4006 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴)
439 cncfmptc 23434 . . . . . . . . . . . . 13 (((𝐹𝑋) ∈ ℝ ∧ ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑧 ∈ ℝ ↦ (𝐹𝑋)) ∈ (ℝ–cn→ℝ))
4408, 154, 154, 439syl3anc 1365 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ ℝ ↦ (𝐹𝑋)) ∈ (ℝ–cn→ℝ))
441 eqidd 2826 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝐹𝑋) = (𝐹𝑋))
442440, 39, 441cnmptlimc 24403 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴))
443438, 442sseldi 3968 . . . . . . . . . 10 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴))
444 limcresi 24398 . . . . . . . . . . . 12 (𝐹 lim 𝐴) ⊆ ((𝐹 ↾ (𝐴(,)𝑋)) lim 𝐴)
4451, 114feqresmpt 6730 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)))
446445oveq1d 7166 . . . . . . . . . . . 12 (𝜑 → ((𝐹 ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
447444, 446sseqtrid 4022 . . . . . . . . . . 11 (𝜑 → (𝐹 lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
448 lhop1.f0 . . . . . . . . . . 11 (𝜑 → 0 ∈ (𝐹 lim 𝐴))
449447, 448sseldd 3971 . . . . . . . . . 10 (𝜑 → 0 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
45046subcn 23389 . . . . . . . . . . 11 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
451 0cn 10625 . . . . . . . . . . . 12 0 ∈ ℂ
452 opelxpi 5590 . . . . . . . . . . . 12 (((𝐹𝑋) ∈ ℂ ∧ 0 ∈ ℂ) → ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ))
4539, 451, 452sylancl 586 . . . . . . . . . . 11 (𝜑 → ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ))
454431toponunii 21440 . . . . . . . . . . . 12 (ℂ × ℂ) = ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld))
455454cncnpi 21802 . . . . . . . . . . 11 (( − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) ∧ ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ)) → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), 0⟩))
456450, 453, 455sylancr 587 . . . . . . . . . 10 (𝜑 → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), 0⟩))
457408, 411, 416, 416, 46, 432, 443, 449, 456limccnp2 24405 . . . . . . . . 9 (𝜑 → ((𝐹𝑋) − 0) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐹𝑋) − (𝐹𝑧))) lim 𝐴))
458429, 457eqeltrrd 2918 . . . . . . . 8 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐹𝑋) − (𝐹𝑧))) lim 𝐴))
45912subid1d 10978 . . . . . . . . 9 (𝜑 → ((𝐺𝑋) − 0) = (𝐺𝑋))
460 limcresi 24398 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴) ⊆ (((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴)
461 resmpt 5903 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)))
462434, 461ax-mp 5 . . . . . . . . . . . . 13 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋))
463462oveq1i 7161 . . . . . . . . . . . 12 (((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴)
464460, 463sseqtri 4006 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴)
465 cncfmptc 23434 . . . . . . . . . . . . 13 (((𝐺𝑋) ∈ ℝ ∧ ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑧 ∈ ℝ ↦ (𝐺𝑋)) ∈ (ℝ–cn→ℝ))
46611, 154, 154, 465syl3anc 1365 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ ℝ ↦ (𝐺𝑋)) ∈ (ℝ–cn→ℝ))
467 eqidd 2826 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝐺𝑋) = (𝐺𝑋))
468466, 39, 467cnmptlimc 24403 . . . . . . . . . . 11 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴))
469464, 468sseldi 3968 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴))
470 limcresi 24398 . . . . . . . . . . . 12 (𝐺 lim 𝐴) ⊆ ((𝐺 ↾ (𝐴(,)𝑋)) lim 𝐴)
47110, 114feqresmpt 6730 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)))
472471oveq1d 7166 . . . . . . . . . . . 12 (𝜑 → ((𝐺 ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
473470, 472sseqtrid 4022 . . . . . . . . . . 11 (𝜑 → (𝐺 lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
474 lhop1.g0 . . . . . . . . . . 11 (𝜑 → 0 ∈ (𝐺 lim 𝐴))
475473, 474sseldd 3971 . . . . . . . . . 10 (𝜑 → 0 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
476 opelxpi 5590 . . . . . . . . . . . 12 (((𝐺𝑋) ∈ ℂ ∧ 0 ∈ ℂ) → ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ))
47712, 451, 476sylancl 586 . . . . . . . . . . 11 (𝜑 → ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ))
478454cncnpi 21802 . . . . . . . . . . 11 (( − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) ∧ ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ)) → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐺𝑋), 0⟩))
479450, 477, 478sylancr 587 . . . . . . . . . 10 (𝜑 → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐺𝑋), 0⟩))
480130, 134, 416, 416, 46, 432, 469, 475, 479limccnp2 24405 . . . . . . . . 9 (𝜑 → ((𝐺𝑋) − 0) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐺𝑋) − (𝐺𝑧))) lim 𝐴))
481459, 480eqeltrrd 2918 . . . . . . . 8 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐺𝑋) − (𝐺𝑧))) lim 𝐴))
482 eqid 2825 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
48346, 482divcn 23391 . . . . . . . . 9 / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld))
484 eldifsn 4717 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ℂ ∖ {0}) ↔ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑋) ≠ 0))
48512, 20, 484sylanbrc 583 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) ∈ (ℂ ∖ {0}))
4869, 485opelxpd 5591 . . . . . . . . 9 (𝜑 → ⟨(𝐹𝑋), (𝐺𝑋)⟩ ∈ (ℂ × (ℂ ∖ {0})))
487 resttopon 21685 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
488419, 417, 487mp2an 688 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0}))
489 txtopon 22115 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0}))) → ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) ∈ (TopOn‘(ℂ × (ℂ ∖ {0}))))
490419, 488, 489mp2an 688 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) ∈ (TopOn‘(ℂ × (ℂ ∖ {0})))
491490toponunii 21440 . . . . . . . . . 10 (ℂ × (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
492491cncnpi 21802 . . . . . . . . 9 (( / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld)) ∧ ⟨(𝐹𝑋), (𝐺𝑋)⟩ ∈ (ℂ × (ℂ ∖ {0}))) → / ∈ ((((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), (𝐺𝑋)⟩))
493483, 486, 492sylancr 587 . . . . . . . 8 (𝜑 → / ∈ ((((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), (𝐺𝑋)⟩))
494412, 415, 416, 418, 46, 428, 458, 481, 493limccnp2 24405 . . . . . . 7 (𝜑 → ((𝐹𝑋) / (𝐺𝑋)) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) lim 𝐴))
495412, 413, 220divcld 11408 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) ∈ ℂ)
496495fmpttd 6874 . . . . . . . 8 (𝜑 → (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))):(𝐴(,)𝑋)⟶ℂ)
497434, 153sstri 3979 . . . . . . . . 9 (𝐴(,)𝑋) ⊆ ℂ
498497a1i 11 . . . . . . . 8 (𝜑 → (𝐴(,)𝑋) ⊆ ℂ)
499496, 498, 66, 46ellimc2 24390 . . . . . . 7 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) lim 𝐴) ↔ (((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢)))))
500494, 499mpbid 233 . . . . . 6 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢))))
501500simprd 496 . . . . 5 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢)))
502 notrab 4283 . . . . . 6 (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}
50368cnmetdval 23294 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝐶𝑥)))
504 abssub 14679 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐶𝑥)) = (abs‘(𝑥𝐶)))
505503, 504eqtrd 2860 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝑥𝐶)))
50624, 505sylan 580 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝑥𝐶)))
507506breq1d 5072 . . . . . . . . 9 ((𝜑𝑥 ∈ ℂ) → ((𝐶(abs ∘ − )𝑥) ≤ 𝐸 ↔ (abs‘(𝑥𝐶)) ≤ 𝐸))
508507rabbidva 3483 . . . . . . . 8 (𝜑 → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} = {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸})
50932a1i 11 . . . . . . . . 9 (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
51028rexrd 10683 . . . . . . . . 9 (𝜑𝐸 ∈ ℝ*)
511 eqid 2825 . . . . . . . . . 10 {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} = {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸}
51247, 511blcld 23030 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝐸 ∈ ℝ*) → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
513509, 24, 510, 512syl3anc 1365 . . . . . . . 8 (𝜑 → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
514508, 513eqeltrrd 2918 . . . . . . 7 (𝜑 → {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
515424cldopn 21555 . . . . . . 7 ({𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)) → (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) ∈ (TopOpen‘ℂfld))
516514, 515syl 17 . . . . . 6 (𝜑 → (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) ∈ (TopOpen‘ℂfld))
517502, 516eqeltrrid 2922 . . . . 5 (𝜑 → {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (TopOpen‘ℂfld))
518407, 501, 517rspcdva 3628 . . . 4 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})))
519402, 518sylbird 261 . . 3 (𝜑 → (¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})))
520397, 519mt3d 150 . 2 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸)
52128recnd 10661 . . . 4 (𝜑𝐸 ∈ ℂ)
522521mulid2d 10651 . . 3 (𝜑 → (1 · 𝐸) = 𝐸)
523 1red 10634 . . . 4 (𝜑 → 1 ∈ ℝ)
524 1lt2 11800 . . . . 5 1 < 2
525524a1i 11 . . . 4 (𝜑 → 1 < 2)
526523, 30, 27, 525ltmul1dd 12479 . . 3 (𝜑 → (1 · 𝐸) < (2 · 𝐸))
527522, 526eqbrtrrd 5086 . 2 (𝜑𝐸 < (2 · 𝐸))
52826, 28, 31, 520, 527lelttrd 10790 1 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) < (2 · 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020  wral 3142  wrex 3143  {crab 3146  Vcvv 3499  cdif 3936  cin 3938  wss 3939  c0 4294  {csn 4563  cop 4569   class class class wbr 5062  cmpt 5142   × cxp 5551  dom cdm 5553  ran crn 5554  cres 5555  cima 5556  ccom 5557   Fn wfn 6346  wf 6347  cfv 6351  (class class class)co 7151  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  *cxr 10666   < clt 10667  cle 10668  cmin 10862   / cdiv 11289  2c2 11684  +crp 12382  (,)cioo 12731  [,]cicc 12734  abscabs 14586  t crest 16686  TopOpenctopn 16687  topGenctg 16703  ∞Metcxmet 20446  ballcbl 20448  fldccnfld 20461  TopOnctopon 21434  Clsdccld 21540  intcnt 21541   Cn ccn 21748   CnP ccnp 21749   ×t ctx 22084  cnccncf 23399   lim climc 24375   D cdv 24376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12383  df-xneg 12500  df-xadd 12501  df-xmul 12502  df-ioo 12735  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-submnd 17947  df-mulg 18157  df-cntz 18379  df-cmn 18830  df-psmet 20453  df-xmet 20454  df-met 20455  df-bl 20456  df-mopn 20457  df-fbas 20458  df-fg 20459  df-cnfld 20462  df-top 21418  df-topon 21435  df-topsp 21457  df-bases 21470  df-cld 21543  df-ntr 21544  df-cls 21545  df-nei 21622  df-lp 21660  df-perf 21661  df-cn 21751  df-cnp 21752  df-haus 21839  df-cmp 21911  df-tx 22086  df-hmeo 22279  df-fil 22370  df-fm 22462  df-flim 22463  df-flf 22464  df-xms 22845  df-ms 22846  df-tms 22847  df-cncf 23401  df-limc 24379  df-dv 24380
This theorem is referenced by:  lhop1  24526
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