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Theorem lhop1lem 23757
 Description: Lemma for lhop1 23758. (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 12196 . . . . . . . . 9 ((𝐵 ∈ ℝ*𝐷𝐵) → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
52, 3, 4syl2anc 692 . . . . . . . 8 (𝜑 → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
6 lhop1lem.x . . . . . . . 8 (𝜑𝑋 ∈ (𝐴(,)𝐷))
75, 6sseldd 3596 . . . . . . 7 (𝜑𝑋 ∈ (𝐴(,)𝐵))
81, 7ffvelrnd 6346 . . . . . 6 (𝜑 → (𝐹𝑋) ∈ ℝ)
98recnd 10053 . . . . 5 (𝜑 → (𝐹𝑋) ∈ ℂ)
10 lhop1.g . . . . . . 7 (𝜑𝐺:(𝐴(,)𝐵)⟶ℝ)
1110, 7ffvelrnd 6346 . . . . . 6 (𝜑 → (𝐺𝑋) ∈ ℝ)
1211recnd 10053 . . . . 5 (𝜑 → (𝐺𝑋) ∈ ℂ)
13 lhop1.gn0 . . . . . 6 (𝜑 → ¬ 0 ∈ ran 𝐺)
14 ffn 6032 . . . . . . . . . 10 (𝐺:(𝐴(,)𝐵)⟶ℝ → 𝐺 Fn (𝐴(,)𝐵))
1510, 14syl 17 . . . . . . . . 9 (𝜑𝐺 Fn (𝐴(,)𝐵))
16 fnfvelrn 6342 . . . . . . . . 9 ((𝐺 Fn (𝐴(,)𝐵) ∧ 𝑋 ∈ (𝐴(,)𝐵)) → (𝐺𝑋) ∈ ran 𝐺)
1715, 7, 16syl2anc 692 . . . . . . . 8 (𝜑 → (𝐺𝑋) ∈ ran 𝐺)
18 eleq1 2687 . . . . . . . 8 ((𝐺𝑋) = 0 → ((𝐺𝑋) ∈ ran 𝐺 ↔ 0 ∈ ran 𝐺))
1917, 18syl5ibcom 235 . . . . . . 7 (𝜑 → ((𝐺𝑋) = 0 → 0 ∈ ran 𝐺))
2019necon3bd 2805 . . . . . 6 (𝜑 → (¬ 0 ∈ ran 𝐺 → (𝐺𝑋) ≠ 0))
2113, 20mpd 15 . . . . 5 (𝜑 → (𝐺𝑋) ≠ 0)
229, 12, 21divcld 10786 . . . 4 (𝜑 → ((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ)
23 limccl 23620 . . . . 5 ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴) ⊆ ℂ
24 lhop1.c . . . . 5 (𝜑𝐶 ∈ ((𝑧 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑧) / ((ℝ D 𝐺)‘𝑧))) lim 𝐴))
2523, 24sseldi 3593 . . . 4 (𝜑𝐶 ∈ ℂ)
2622, 25subcld 10377 . . 3 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) − 𝐶) ∈ ℂ)
2726abscld 14156 . 2 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ∈ ℝ)
28 lhop1lem.e . . 3 (𝜑𝐸 ∈ ℝ+)
2928rpred 11857 . 2 (𝜑𝐸 ∈ ℝ)
30 2re 11075 . . . 4 2 ∈ ℝ
3130a1i 11 . . 3 (𝜑 → 2 ∈ ℝ)
3231, 29remulcld 10055 . 2 (𝜑 → (2 · 𝐸) ∈ ℝ)
33 cnxmet 22557 . . . . . . . . . . . . 13 (abs ∘ − ) ∈ (∞Met‘ℂ)
3433a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
35 simprl 793 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → 𝑣 ∈ (TopOpen‘ℂfld))
36 simprr 795 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → 𝐴𝑣)
37 eliooord 12218 . . . . . . . . . . . . . . . 16 (𝑋 ∈ (𝐴(,)𝐷) → (𝐴 < 𝑋𝑋 < 𝐷))
386, 37syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 < 𝑋𝑋 < 𝐷))
3938simpld 475 . . . . . . . . . . . . . 14 (𝜑𝐴 < 𝑋)
40 lhop1.a . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ ℝ)
41 ioossre 12220 . . . . . . . . . . . . . . . 16 (𝐴(,)𝐷) ⊆ ℝ
4241, 6sseldi 3593 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ ℝ)
43 difrp 11853 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (𝐴 < 𝑋 ↔ (𝑋𝐴) ∈ ℝ+))
4440, 42, 43syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → (𝐴 < 𝑋 ↔ (𝑋𝐴) ∈ ℝ+))
4539, 44mpbid 222 . . . . . . . . . . . . 13 (𝜑 → (𝑋𝐴) ∈ ℝ+)
4645adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → (𝑋𝐴) ∈ ℝ+)
47 eqid 2620 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4847cnfldtopn 22566 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) = (MetOpen‘(abs ∘ − ))
4948mopni3 22280 . . . . . . . . . . . 12 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣) ∧ (𝑋𝐴) ∈ ℝ+) → ∃𝑟 ∈ ℝ+ (𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣))
5034, 35, 36, 46, 49syl31anc 1327 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑟 ∈ ℝ+ (𝑟 < (𝑋𝐴) ∧ (𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣))
51 lhop1lem.r . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 = (𝐴 + (𝑟 / 2))
5240adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℝ)
53 simprl 793 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ+)
5453rpred 11857 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ)
5554rehalfcld 11264 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ)
5652, 55readdcld 10054 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴 + (𝑟 / 2)) ∈ ℝ)
5751, 56syl5eqel 2703 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℝ)
5857recnd 10053 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℂ)
5940recnd 10053 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ ℂ)
6059adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℂ)
61 eqid 2620 . . . . . . . . . . . . . . . . . . . . . . 23 (abs ∘ − ) = (abs ∘ − )
6261cnmetdval 22555 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑅(abs ∘ − )𝐴) = (abs‘(𝑅𝐴)))
6358, 60, 62syl2anc 692 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) = (abs‘(𝑅𝐴)))
6451oveq1i 6645 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑅𝐴) = ((𝐴 + (𝑟 / 2)) − 𝐴)
6554recnd 10053 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℂ)
6665halfcld 11262 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℂ)
6760, 66pncan2d 10379 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐴 + (𝑟 / 2)) − 𝐴) = (𝑟 / 2))
6864, 67syl5eq 2666 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅𝐴) = (𝑟 / 2))
6968fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘(𝑅𝐴)) = (abs‘(𝑟 / 2)))
7053rphalfcld 11869 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ+)
7170rpred 11857 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) ∈ ℝ)
7270rpge0d 11861 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 0 ≤ (𝑟 / 2))
7371, 72absidd 14142 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘(𝑟 / 2)) = (𝑟 / 2))
7463, 69, 733eqtrd 2658 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) = (𝑟 / 2))
75 rphalflt 11845 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ+ → (𝑟 / 2) < 𝑟)
7653, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) < 𝑟)
7774, 76eqbrtrd 4666 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(abs ∘ − )𝐴) < 𝑟)
7833a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs ∘ − ) ∈ (∞Met‘ℂ))
7954rexrd 10074 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 ∈ ℝ*)
80 elbl3 22178 . . . . . . . . . . . . . . . . . . . 20 ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑟 ∈ ℝ*) ∧ (𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ)) → (𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐴) < 𝑟))
8178, 79, 60, 58, 80syl22anc 1325 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟) ↔ (𝑅(abs ∘ − )𝐴) < 𝑟))
8277, 81mpbird 247 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(ball‘(abs ∘ − ))𝑟))
8352, 70ltaddrpd 11890 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 < (𝐴 + (𝑟 / 2)))
8483, 51syl6breqr 4686 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 < 𝑅)
8542adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 ∈ ℝ)
8685, 52resubcld 10443 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑋𝐴) ∈ ℝ)
87 simprr 795 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑟 < (𝑋𝐴))
8871, 54, 86, 76, 87lttrd 10183 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑟 / 2) < (𝑋𝐴))
8952, 71, 85ltaddsub2d 10613 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐴 + (𝑟 / 2)) < 𝑋 ↔ (𝑟 / 2) < (𝑋𝐴)))
9088, 89mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴 + (𝑟 / 2)) < 𝑋)
9151, 90syl5eqbr 4679 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 < 𝑋)
9252rexrd 10074 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴 ∈ ℝ*)
9342rexrd 10074 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑋 ∈ ℝ*)
9493adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 ∈ ℝ*)
95 elioo2 12201 . . . . . . . . . . . . . . . . . . . 20 ((𝐴 ∈ ℝ*𝑋 ∈ ℝ*) → (𝑅 ∈ (𝐴(,)𝑋) ↔ (𝑅 ∈ ℝ ∧ 𝐴 < 𝑅𝑅 < 𝑋)))
9692, 94, 95syl2anc 692 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅 ∈ (𝐴(,)𝑋) ↔ (𝑅 ∈ ℝ ∧ 𝐴 < 𝑅𝑅 < 𝑋)))
9757, 84, 91, 96mpbir3and 1243 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(,)𝑋))
9882, 97elind 3790 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)))
999adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑋) ∈ ℂ)
1001adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
101 lhop1lem.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝐷 ∈ ℝ)
102101rexrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐷 ∈ ℝ*)
10338simprd 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑋 < 𝐷)
10442, 101, 103ltled 10170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑋𝐷)
10593, 102, 2, 104, 3xrletrd 11978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑋𝐵)
106 iooss2 12196 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ ℝ*𝑋𝐵) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
1072, 105, 106syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
108107adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐵))
109108, 97sseldd 3596 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ (𝐴(,)𝐵))
110100, 109ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑅) ∈ ℝ)
111110recnd 10053 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹𝑅) ∈ ℂ)
11299, 111subcld 10377 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐹𝑋) − (𝐹𝑅)) ∈ ℂ)
11312adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑋) ∈ ℂ)
11410adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
115114, 109ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑅) ∈ ℝ)
116115recnd 10053 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺𝑅) ∈ ℂ)
117113, 116subcld 10377 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐺𝑋) − (𝐺𝑅)) ∈ ℂ)
118 lhop1.gd0 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ¬ 0 ∈ ran (ℝ D 𝐺))
119118adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐺))
12012adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑋) ∈ ℂ)
121107sselda 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ (𝐴(,)𝐵))
12210ffvelrnda 6345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑧 ∈ (𝐴(,)𝐵)) → (𝐺𝑧) ∈ ℝ)
123121, 122syldan 487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑧) ∈ ℝ)
124123recnd 10053 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐺𝑧) ∈ ℂ)
125120, 124subeq0ad 10387 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑧)) = 0 ↔ (𝐺𝑋) = (𝐺𝑧)))
126 ioossre 12220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝐴(,)𝐵) ⊆ ℝ
127126, 121sseldi 3593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ ℝ)
128127adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑧 ∈ ℝ)
12942ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑋 ∈ ℝ)
130 eliooord 12218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ (𝐴(,)𝑋) → (𝐴 < 𝑧𝑧 < 𝑋))
131130adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐴 < 𝑧𝑧 < 𝑋))
132131simprd 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 < 𝑋)
133132adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝑧 < 𝑋)
13440rexrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝐴 ∈ ℝ*)
135134adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐴 ∈ ℝ*)
1362adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐵 ∈ ℝ*)
137131simpld 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝐴 < 𝑧)
13893, 102, 2, 103, 3xrltletrd 11977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝑋 < 𝐵)
139138adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 < 𝐵)
140 iccssioo 12227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ (𝐴 < 𝑧𝑋 < 𝐵)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
141135, 136, 137, 139, 140syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
142141adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧[,]𝑋) ⊆ (𝐴(,)𝐵))
143 ax-resscn 9978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ℝ ⊆ ℂ
144143a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → ℝ ⊆ ℂ)
145 fss 6043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
14610, 143, 145sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑𝐺:(𝐴(,)𝐵)⟶ℂ)
147126a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
148 lhop1.ig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
149 dvcn 23665 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((ℝ ⊆ ℂ ∧ 𝐺:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
150144, 146, 147, 148, 149syl31anc 1327 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
151 cncffvrn 22682 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((ℝ ⊆ ℂ ∧ 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐺:(𝐴(,)𝐵)⟶ℝ))
152143, 150, 151sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐺:(𝐴(,)𝐵)⟶ℝ))
15310, 152mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
154153ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
155 rescncf 22681 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑧[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐺 ↾ (𝑧[,]𝑋)) ∈ ((𝑧[,]𝑋)–cn→ℝ)))
156142, 154, 155sylc 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝐺 ↾ (𝑧[,]𝑋)) ∈ ((𝑧[,]𝑋)–cn→ℝ))
157143a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ℝ ⊆ ℂ)
158146ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
159126a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝐴(,)𝐵) ⊆ ℝ)
160142, 126syl6ss 3607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧[,]𝑋) ⊆ ℝ)
16147tgioo2 22587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
16247, 161dvres 23656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
163157, 158, 159, 160, 162syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋))))
164 iccntr 22605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋)) = (𝑧(,)𝑋))
165128, 129, 164syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋)) = (𝑧(,)𝑋))
166165reseq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
167163, 166eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
168167dmeqd 5315 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)))
169 ioossicc 12244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧(,)𝑋) ⊆ (𝑧[,]𝑋)
170169, 142syl5ss 3606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧(,)𝑋) ⊆ (𝐴(,)𝐵))
171148ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
172170, 171sseqtr4d 3634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (𝑧(,)𝑋) ⊆ dom (ℝ D 𝐺))
173 ssdmres 5408 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑧(,)𝑋) ⊆ dom (ℝ D 𝐺) ↔ dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)) = (𝑧(,)𝑋))
174172, 173sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom ((ℝ D 𝐺) ↾ (𝑧(,)𝑋)) = (𝑧(,)𝑋))
175168, 174eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → dom (ℝ D (𝐺 ↾ (𝑧[,]𝑋))) = (𝑧(,)𝑋))
176127rexrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ ℝ*)
17793adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ ℝ*)
17842adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ ℝ)
179127, 178, 132ltled 10170 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧𝑋)
180 ubicc2 12274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ*𝑋 ∈ ℝ*𝑧𝑋) → 𝑋 ∈ (𝑧[,]𝑋))
181176, 177, 179, 180syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑋 ∈ (𝑧[,]𝑋))
182 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑋 ∈ (𝑧[,]𝑋) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = (𝐺𝑋))
183181, 182syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = (𝐺𝑋))
184 lbicc2 12273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑧 ∈ ℝ*𝑋 ∈ ℝ*𝑧𝑋) → 𝑧 ∈ (𝑧[,]𝑋))
185176, 177, 179, 184syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → 𝑧 ∈ (𝑧[,]𝑋))
186 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 ∈ (𝑧[,]𝑋) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) = (𝐺𝑧))
187185, 186syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) = (𝐺𝑧))
188183, 187eqeq12d 2635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) ↔ (𝐺𝑋) = (𝐺𝑧)))
189188biimpar 502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧))
190189eqcomd 2626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((𝐺 ↾ (𝑧[,]𝑋))‘𝑧) = ((𝐺 ↾ (𝑧[,]𝑋))‘𝑋))
191128, 129, 133, 156, 175, 190rolle 23734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ∃𝑤 ∈ (𝑧(,)𝑋)((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0)
192167fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = (((ℝ D 𝐺) ↾ (𝑧(,)𝑋))‘𝑤))
193 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑤 ∈ (𝑧(,)𝑋) → (((ℝ D 𝐺) ↾ (𝑧(,)𝑋))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
194192, 193sylan9eq 2674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
195 dvf 23652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
196148feq2d 6018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝜑 → ((ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ ↔ (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ))
197195, 196mpbii 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑 → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
198197ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
199 ffn 6032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
200198, 199syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
201200adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
202170sselda 3595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐵))
203 fnfvelrn 6342 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((ℝ D 𝐺) Fn (𝐴(,)𝐵) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
204201, 202, 203syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
205194, 204eqeltrd 2699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) ∈ ran (ℝ D 𝐺))
206 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
207205, 206syl5ibcom 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) ∧ 𝑤 ∈ (𝑧(,)𝑋)) → (((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
208207rexlimdva 3027 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → (∃𝑤 ∈ (𝑧(,)𝑋)((ℝ D (𝐺 ↾ (𝑧[,]𝑋)))‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
209191, 208mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑧 ∈ (𝐴(,)𝑋)) ∧ (𝐺𝑋) = (𝐺𝑧)) → 0 ∈ ran (ℝ D 𝐺))
210209ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) = (𝐺𝑧) → 0 ∈ ran (ℝ D 𝐺)))
211125, 210sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑧)) = 0 → 0 ∈ ran (ℝ D 𝐺)))
212211necon3bd 2805 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((𝐺𝑋) − (𝐺𝑧)) ≠ 0))
213119, 212mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
214213ralrimiva 2963 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑧 ∈ (𝐴(,)𝑋)((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
215214adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∀𝑧 ∈ (𝐴(,)𝑋)((𝐺𝑋) − (𝐺𝑧)) ≠ 0)
216 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = 𝑅 → (𝐺𝑧) = (𝐺𝑅))
217216oveq2d 6651 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 = 𝑅 → ((𝐺𝑋) − (𝐺𝑧)) = ((𝐺𝑋) − (𝐺𝑅)))
218217neeq1d 2850 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = 𝑅 → (((𝐺𝑋) − (𝐺𝑧)) ≠ 0 ↔ ((𝐺𝑋) − (𝐺𝑅)) ≠ 0))
219218rspcv 3300 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ (𝐴(,)𝑋) → (∀𝑧 ∈ (𝐴(,)𝑋)((𝐺𝑋) − (𝐺𝑧)) ≠ 0 → ((𝐺𝑋) − (𝐺𝑅)) ≠ 0))
22097, 215, 219sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((𝐺𝑋) − (𝐺𝑅)) ≠ 0)
221112, 117, 220divcld 10786 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) ∈ ℂ)
22225adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐶 ∈ ℂ)
223221, 222subcld 10377 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶) ∈ ℂ)
224223abscld 14156 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ∈ ℝ)
22529adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐸 ∈ ℝ)
226102adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐷 ∈ ℝ*)
227103adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋 < 𝐷)
228 iccssioo 12227 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐴 ∈ ℝ*𝐷 ∈ ℝ*) ∧ (𝐴 < 𝑅𝑋 < 𝐷)) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐷))
22992, 226, 84, 227, 228syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐷))
2305adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝐷) ⊆ (𝐴(,)𝐵))
231229, 230sstrd 3605 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ (𝐴(,)𝐵))
232 fss 6043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
2331, 143, 232sylancl 693 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝐹:(𝐴(,)𝐵)⟶ℂ)
234 lhop1.if . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
235 dvcn 23665 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
236144, 233, 147, 234, 235syl31anc 1327 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
237 cncffvrn 22682 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ℝ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
238143, 236, 237sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ))
2391, 238mpbird 247 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
240239adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
241 rescncf 22681 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐹 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ)))
242231, 240, 241sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐹 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ))
243153adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ))
244 rescncf 22681 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅[,]𝑋) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℝ) → (𝐺 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ)))
245231, 243, 244sylc 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐺 ↾ (𝑅[,]𝑋)) ∈ ((𝑅[,]𝑋)–cn→ℝ))
246143a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ℝ ⊆ ℂ)
247233adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
248126a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝐵) ⊆ ℝ)
249 iccssre 12240 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ ℝ ∧ 𝑋 ∈ ℝ) → (𝑅[,]𝑋) ⊆ ℝ)
25057, 85, 249syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅[,]𝑋) ⊆ ℝ)
25147, 161dvres 23656 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
252246, 247, 248, 250, 251syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
253 iccntr 22605 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑅 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋)) = (𝑅(,)𝑋))
25457, 85, 253syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋)) = (𝑅(,)𝑋))
255254reseq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D 𝐹) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
256252, 255eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
257256dmeqd 5315 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)))
25852, 57, 84ltled 10170 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐴𝑅)
259 iooss1 12195 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐴 ∈ ℝ*𝐴𝑅) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝑋))
26092, 258, 259syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝑋))
261104adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑋𝐷)
262 iooss2 12196 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐷 ∈ ℝ*𝑋𝐷) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐷))
263226, 261, 262syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝐴(,)𝑋) ⊆ (𝐴(,)𝐷))
264260, 263sstrd 3605 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝐷))
265264, 230sstrd 3605 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ (𝐴(,)𝐵))
266234adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
267265, 266sseqtr4d 3634 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ dom (ℝ D 𝐹))
268 ssdmres 5408 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅(,)𝑋) ⊆ dom (ℝ D 𝐹) ↔ dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
269267, 268sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom ((ℝ D 𝐹) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
270257, 269eqtrd 2654 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐹 ↾ (𝑅[,]𝑋))) = (𝑅(,)𝑋))
271146adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
27247, 161dvres 23656 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
273246, 271, 248, 250, 272syl22anc 1325 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))))
274254reseq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D 𝐺) ↾ ((int‘(topGen‘ran (,)))‘(𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
275273, 274eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
276275dmeqd 5315 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)))
277148adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
278265, 277sseqtr4d 3634 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (𝑅(,)𝑋) ⊆ dom (ℝ D 𝐺))
279 ssdmres 5408 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅(,)𝑋) ⊆ dom (ℝ D 𝐺) ↔ dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
280278, 279sylib 208 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom ((ℝ D 𝐺) ↾ (𝑅(,)𝑋)) = (𝑅(,)𝑋))
281276, 280eqtrd 2654 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → dom (ℝ D (𝐺 ↾ (𝑅[,]𝑋))) = (𝑅(,)𝑋))
28257, 85, 91, 242, 245, 270, 281cmvth 23735 . . . . . . . . . . . . . . . . . . . 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)))
28357rexrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅 ∈ ℝ*)
284283adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅 ∈ ℝ*)
28593ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑋 ∈ ℝ*)
28657, 85, 91ltled 10170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → 𝑅𝑋)
287286adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅𝑋)
288 ubicc2 12274 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ ℝ*𝑋 ∈ ℝ*𝑅𝑋) → 𝑋 ∈ (𝑅[,]𝑋))
289284, 285, 287, 288syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑋 ∈ (𝑅[,]𝑋))
290 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑋 ∈ (𝑅[,]𝑋) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐹𝑋))
291289, 290syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐹𝑋))
292 lbicc2 12273 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑅 ∈ ℝ*𝑋 ∈ ℝ*𝑅𝑋) → 𝑅 ∈ (𝑅[,]𝑋))
293284, 285, 287, 292syl3anc 1324 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑅 ∈ (𝑅[,]𝑋))
294 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑅 ∈ (𝑅[,]𝑋) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐹𝑅))
295293, 294syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐹𝑅))
296291, 295oveq12d 6653 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐹 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐹 ↾ (𝑅[,]𝑋))‘𝑅)) = ((𝐹𝑋) − (𝐹𝑅)))
297275fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤) = (((ℝ D 𝐺) ↾ (𝑅(,)𝑋))‘𝑤))
298 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 ∈ (𝑅(,)𝑋) → (((ℝ D 𝐺) ↾ (𝑅(,)𝑋))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
299297, 298sylan9eq 2674 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D (𝐺 ↾ (𝑅[,]𝑋)))‘𝑤) = ((ℝ D 𝐺)‘𝑤))
300296, 299oveq12d 6653 . . . . . . . . . . . . . . . . . . . . . . 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)))
301 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑋 ∈ (𝑅[,]𝑋) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐺𝑋))
302289, 301syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) = (𝐺𝑋))
303 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑅 ∈ (𝑅[,]𝑋) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐺𝑅))
304293, 303syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅) = (𝐺𝑅))
305302, 304oveq12d 6653 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐺 ↾ (𝑅[,]𝑋))‘𝑋) − ((𝐺 ↾ (𝑅[,]𝑋))‘𝑅)) = ((𝐺𝑋) − (𝐺𝑅)))
306256fveq1d 6180 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤) = (((ℝ D 𝐹) ↾ (𝑅(,)𝑋))‘𝑤))
307 fvres 6194 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑅(,)𝑋) → (((ℝ D 𝐹) ↾ (𝑅(,)𝑋))‘𝑤) = ((ℝ D 𝐹)‘𝑤))
308306, 307sylan9eq 2674 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D (𝐹 ↾ (𝑅[,]𝑋)))‘𝑤) = ((ℝ D 𝐹)‘𝑤))
309305, 308oveq12d 6653 . . . . . . . . . . . . . . . . . . . . . . . 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)))
310117adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑅)) ∈ ℂ)
311 dvf 23652 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
312234feq2d 6018 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
313311, 312mpbii 223 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
314313ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
315265sselda 3595 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐵))
316314, 315ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐹)‘𝑤) ∈ ℂ)
317310, 316mulcomd 10046 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((𝐺𝑋) − (𝐺𝑅)) · ((ℝ D 𝐹)‘𝑤)) = (((ℝ D 𝐹)‘𝑤) · ((𝐺𝑋) − (𝐺𝑅))))
318309, 317eqtrd 2654 . . . . . . . . . . . . . . . . . . . . . . 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))))
319300, 318eqeq12d 2635 . . . . . . . . . . . . . . . . . . . . . 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)))))
320112adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐹𝑋) − (𝐹𝑅)) ∈ ℂ)
321197ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐺):(𝐴(,)𝐵)⟶ℂ)
322321, 315ffvelrnd 6346 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ℂ)
323220adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑅)) ≠ 0)
324118ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ¬ 0 ∈ ran (ℝ D 𝐺))
325321, 199syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
326325, 315, 203syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺))
327 eleq1 2687 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((ℝ D 𝐺)‘𝑤) = 0 → (((ℝ D 𝐺)‘𝑤) ∈ ran (ℝ D 𝐺) ↔ 0 ∈ ran (ℝ D 𝐺)))
328326, 327syl5ibcom 235 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (((ℝ D 𝐺)‘𝑤) = 0 → 0 ∈ ran (ℝ D 𝐺)))
329328necon3bd 2805 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (¬ 0 ∈ ran (ℝ D 𝐺) → ((ℝ D 𝐺)‘𝑤) ≠ 0))
330324, 329mpd 15 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ((ℝ D 𝐺)‘𝑤) ≠ 0)
331320, 310, 316, 322, 323, 330divmuleqd 10832 . . . . . . . . . . . . . . . . . . . . . 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)))))
332319, 331bitr4d 271 . . . . . . . . . . . . . . . . . . . . 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))))
333332rexbidva 3045 . . . . . . . . . . . . . . . . . . . 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))))
334282, 333mpbid 222 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑤 ∈ (𝑅(,)𝑋)(((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)))
335264sselda 3595 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → 𝑤 ∈ (𝐴(,)𝐷))
336 lhop1lem.t . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)
337336ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → ∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸)
338 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑤 → ((ℝ D 𝐹)‘𝑡) = ((ℝ D 𝐹)‘𝑤))
339 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑡 = 𝑤 → ((ℝ D 𝐺)‘𝑡) = ((ℝ D 𝐺)‘𝑤))
340338, 339oveq12d 6653 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑡 = 𝑤 → (((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)))
341340oveq1d 6650 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑡 = 𝑤 → ((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶) = ((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶))
342341fveq2d 6182 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 = 𝑤 → (abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) = (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)))
343342breq1d 4654 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 = 𝑤 → ((abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸 ↔ (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸))
344343rspcv 3300 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 ∈ (𝐴(,)𝐷) → (∀𝑡 ∈ (𝐴(,)𝐷)(abs‘((((ℝ D 𝐹)‘𝑡) / ((ℝ D 𝐺)‘𝑡)) − 𝐶)) < 𝐸 → (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸))
345335, 337, 344sylc 65 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) ∧ 𝑤 ∈ (𝑅(,)𝑋)) → (abs‘((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶)) < 𝐸)
346 oveq1 6642 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) = (((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) → ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶) = ((((ℝ D 𝐹)‘𝑤) / ((ℝ D 𝐺)‘𝑤)) − 𝐶))
347346fveq2d 6182 . . . . . . . . . . . . . . . . . . . . . 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)))
348347breq1d 4654 . . . . . . . . . . . . . . . . . . . . 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))
349345, 348syl5ibrcom 237 . . . . . . . . . . . . . . . . . . . 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))
350349rexlimdva 3027 . . . . . . . . . . . . . . . . . . 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))
351334, 350mpd 15 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) < 𝐸)
352224, 225, 351ltled 10170 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ≤ 𝐸)
353 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑅 → (𝐹𝑢) = (𝐹𝑅))
354353oveq2d 6651 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑅 → ((𝐹𝑋) − (𝐹𝑢)) = ((𝐹𝑋) − (𝐹𝑅)))
355 fveq2 6178 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑅 → (𝐺𝑢) = (𝐺𝑅))
356355oveq2d 6651 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑅 → ((𝐺𝑋) − (𝐺𝑢)) = ((𝐺𝑋) − (𝐺𝑅)))
357354, 356oveq12d 6653 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑅 → (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) = (((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))))
358357oveq1d 6650 . . . . . . . . . . . . . . . . . . . 20 (𝑢 = 𝑅 → ((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶) = ((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶))
359358fveq2d 6182 . . . . . . . . . . . . . . . . . . 19 (𝑢 = 𝑅 → (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) = (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)))
360359breq1d 4654 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑅 → ((abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸 ↔ (abs‘((((𝐹𝑋) − (𝐹𝑅)) / ((𝐺𝑋) − (𝐺𝑅))) − 𝐶)) ≤ 𝐸))
361360rspcev 3304 . . . . . . . . . . . . . . . . 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)
36298, 352, 361syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
363362adantlr 750 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) ∧ (𝑟 ∈ ℝ+𝑟 < (𝑋𝐴))) → ∃𝑢 ∈ ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
364 ssrin 3830 . . . . . . . . . . . . . . . . 17 ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ⊆ (𝑣 ∩ (𝐴(,)𝑋)))
365 lbioo 12191 . . . . . . . . . . . . . . . . . . . 20 ¬ 𝐴 ∈ (𝐴(,)𝑋)
366 disjsn 4237 . . . . . . . . . . . . . . . . . . . 20 (((𝐴(,)𝑋) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ (𝐴(,)𝑋))
367365, 366mpbir 221 . . . . . . . . . . . . . . . . . . 19 ((𝐴(,)𝑋) ∩ {𝐴}) = ∅
368 disj3 4012 . . . . . . . . . . . . . . . . . . 19 (((𝐴(,)𝑋) ∩ {𝐴}) = ∅ ↔ (𝐴(,)𝑋) = ((𝐴(,)𝑋) ∖ {𝐴}))
369367, 368mpbi 220 . . . . . . . . . . . . . . . . . 18 (𝐴(,)𝑋) = ((𝐴(,)𝑋) ∖ {𝐴})
370369ineq2i 3803 . . . . . . . . . . . . . . . . 17 (𝑣 ∩ (𝐴(,)𝑋)) = (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))
371364, 370syl6sseq 3643 . . . . . . . . . . . . . . . 16 ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → ((𝐴(ball‘(abs ∘ − ))𝑟) ∩ (𝐴(,)𝑋)) ⊆ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))
372 ssrexv 3659 . . . . . . . . . . . . . . . 16 (((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))
373371, 372syl 17 . . . . . . . . . . . . . . 15 ((𝐴(ball‘(abs ∘ − ))𝑟) ⊆ 𝑣 → (∃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))
374363, 373syl5com 31 . . . . . . . . . . . . . 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))
375374anassrs 679 . . . . . . . . . . . . 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))
376375expimpd 628 . . . . . . . . . . . 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))
377376rexlimdva 3027 . . . . . . . . . . 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))
37850, 377mpd 15 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸)
379 inss2 3826 . . . . . . . . . . . . . 14 (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) ⊆ ((𝐴(,)𝑋) ∖ {𝐴})
380 difss 3729 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ∖ {𝐴}) ⊆ (𝐴(,)𝑋)
381379, 380sstri 3604 . . . . . . . . . . . . 13 (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) ⊆ (𝐴(,)𝑋)
382381sseli 3591 . . . . . . . . . . . 12 (𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})) → 𝑢 ∈ (𝐴(,)𝑋))
383 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑢 → (𝐹𝑧) = (𝐹𝑢))
384383oveq2d 6651 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → ((𝐹𝑋) − (𝐹𝑧)) = ((𝐹𝑋) − (𝐹𝑢)))
385 fveq2 6178 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑢 → (𝐺𝑧) = (𝐺𝑢))
386385oveq2d 6651 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑢 → ((𝐺𝑋) − (𝐺𝑧)) = ((𝐺𝑋) − (𝐺𝑢)))
387384, 386oveq12d 6653 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑢 → (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) = (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))))
388 eqid 2620 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))
389 ovex 6663 . . . . . . . . . . . . . . . 16 (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) ∈ V
390387, 388, 389fvmpt 6269 . . . . . . . . . . . . . . 15 (𝑢 ∈ (𝐴(,)𝑋) → ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) = (((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))))
391390oveq1d 6650 . . . . . . . . . . . . . 14 (𝑢 ∈ (𝐴(,)𝑋) → (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶) = ((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶))
392391fveq2d 6182 . . . . . . . . . . . . 13 (𝑢 ∈ (𝐴(,)𝑋) → (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) = (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)))
393392breq1d 4654 . . . . . . . . . . . 12 (𝑢 ∈ (𝐴(,)𝑋) → ((abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸 ↔ (abs‘((((𝐹𝑋) − (𝐹𝑢)) / ((𝐺𝑋) − (𝐺𝑢))) − 𝐶)) ≤ 𝐸))
394382, 393syl 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))
395394rexbiia 3036 . . . . . . . . . 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)
396378, 395sylibr 224 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑢 ∈ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))(abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸)
397 ovex 6663 . . . . . . . . . . 11 (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) ∈ V
398397, 388fnmpti 6009 . . . . . . . . . 10 (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) Fn (𝐴(,)𝑋)
399 oveq1 6642 . . . . . . . . . . . . 13 (𝑥 = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) → (𝑥𝐶) = (((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶))
400399fveq2d 6182 . . . . . . . . . . . 12 (𝑥 = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) → (abs‘(𝑥𝐶)) = (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)))
401400breq1d 4654 . . . . . . . . . . 11 (𝑥 = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) → ((abs‘(𝑥𝐶)) ≤ 𝐸 ↔ (abs‘(((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))))‘𝑢) − 𝐶)) ≤ 𝐸))
402401rexima 6482 . . . . . . . . . 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))
403398, 381, 402mp2an 707 . . . . . . . . 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)
404396, 403sylibr 224 . . . . . . . 8 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ∃𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴})))(abs‘(𝑥𝐶)) ≤ 𝐸)
405 dfrex2 2993 . . . . . . . 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)
406404, 405sylib 208 . . . . . . 7 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ¬ ∀𝑥 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ¬ (abs‘(𝑥𝐶)) ≤ 𝐸)
407 ssrab 3672 . . . . . . . 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))
408407simprbi 480 . . . . . . 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)
409406, 408nsyl 135 . . . . . 6 ((𝜑 ∧ (𝑣 ∈ (TopOpen‘ℂfld) ∧ 𝐴𝑣)) → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸})
410409expr 642 . . . . 5 ((𝜑𝑣 ∈ (TopOpen‘ℂfld)) → (𝐴𝑣 → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
411410ralrimiva 2963 . . . 4 (𝜑 → ∀𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 → ¬ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
412 ralinexa 2994 . . . 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}))
413411, 412sylib 208 . . 3 (𝜑 → ¬ ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
414 oveq1 6642 . . . . . . . . 9 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → (𝑥𝐶) = (((𝐹𝑋) / (𝐺𝑋)) − 𝐶))
415414fveq2d 6182 . . . . . . . 8 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → (abs‘(𝑥𝐶)) = (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)))
416415breq1d 4654 . . . . . . 7 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → ((abs‘(𝑥𝐶)) ≤ 𝐸 ↔ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
417416notbid 308 . . . . . 6 (𝑥 = ((𝐹𝑋) / (𝐺𝑋)) → (¬ (abs‘(𝑥𝐶)) ≤ 𝐸 ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
418417elrab3 3358 . . . . 5 (((𝐹𝑋) / (𝐺𝑋)) ∈ ℂ → (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
41922, 418syl 17 . . . 4 (𝜑 → (((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ↔ ¬ (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸))
420 notrab 3896 . . . . . 6 (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}
42161cnmetdval 22555 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝐶𝑥)))
422 abssub 14047 . . . . . . . . . . . 12 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (abs‘(𝐶𝑥)) = (abs‘(𝑥𝐶)))
423421, 422eqtrd 2654 . . . . . . . . . . 11 ((𝐶 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝑥𝐶)))
42425, 423sylan 488 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℂ) → (𝐶(abs ∘ − )𝑥) = (abs‘(𝑥𝐶)))
425424breq1d 4654 . . . . . . . . 9 ((𝜑𝑥 ∈ ℂ) → ((𝐶(abs ∘ − )𝑥) ≤ 𝐸 ↔ (abs‘(𝑥𝐶)) ≤ 𝐸))
426425rabbidva 3183 . . . . . . . 8 (𝜑 → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} = {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸})
42733a1i 11 . . . . . . . . 9 (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
42829rexrd 10074 . . . . . . . . 9 (𝜑𝐸 ∈ ℝ*)
429 eqid 2620 . . . . . . . . . 10 {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} = {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸}
43048, 429blcld 22291 . . . . . . . . 9 (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐶 ∈ ℂ ∧ 𝐸 ∈ ℝ*) → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
431427, 25, 428, 430syl3anc 1324 . . . . . . . 8 (𝜑 → {𝑥 ∈ ℂ ∣ (𝐶(abs ∘ − )𝑥) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
432426, 431eqeltrrd 2700 . . . . . . 7 (𝜑 → {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)))
43347cnfldtopon 22567 . . . . . . . . 9 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
434433toponunii 20702 . . . . . . . 8 ℂ = (TopOpen‘ℂfld)
435434cldopn 20816 . . . . . . 7 ({𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (Clsd‘(TopOpen‘ℂfld)) → (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) ∈ (TopOpen‘ℂfld))
436432, 435syl 17 . . . . . 6 (𝜑 → (ℂ ∖ {𝑥 ∈ ℂ ∣ (abs‘(𝑥𝐶)) ≤ 𝐸}) ∈ (TopOpen‘ℂfld))
437420, 436syl5eqelr 2704 . . . . 5 (𝜑 → {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (TopOpen‘ℂfld))
4389adantr 481 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑋) ∈ ℂ)
4391ffvelrnda 6345 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝐴(,)𝐵)) → (𝐹𝑧) ∈ ℝ)
440121, 439syldan 487 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑧) ∈ ℝ)
441440recnd 10053 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (𝐹𝑧) ∈ ℂ)
442438, 441subcld 10377 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐹𝑋) − (𝐹𝑧)) ∈ ℂ)
443120, 124subcld 10377 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ∈ ℂ)
444 eldifsn 4308 . . . . . . . . 9 (((𝐺𝑋) − (𝐺𝑧)) ∈ (ℂ ∖ {0}) ↔ (((𝐺𝑋) − (𝐺𝑧)) ∈ ℂ ∧ ((𝐺𝑋) − (𝐺𝑧)) ≠ 0))
445443, 213, 444sylanbrc 697 . . . . . . . 8 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → ((𝐺𝑋) − (𝐺𝑧)) ∈ (ℂ ∖ {0}))
446 ssid 3616 . . . . . . . . 9 ℂ ⊆ ℂ
447446a1i 11 . . . . . . . 8 (𝜑 → ℂ ⊆ ℂ)
448 difss 3729 . . . . . . . . 9 (ℂ ∖ {0}) ⊆ ℂ
449448a1i 11 . . . . . . . 8 (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
450 cnex 10002 . . . . . . . . . 10 ℂ ∈ V
451450, 448ssexi 4794 . . . . . . . . . 10 (ℂ ∖ {0}) ∈ V
452 txrest 21415 . . . . . . . . . 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}))))
453433, 433, 450, 451, 452mp4an 708 . . . . . . . . 9 (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × (ℂ ∖ {0}))) = (((TopOpen‘ℂfld) ↾t ℂ) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
454434restid 16075 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
455433, 454ax-mp 5 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
456455oveq1i 6645 . . . . . . . . 9 (((TopOpen‘ℂfld) ↾t ℂ) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) = ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
457453, 456eqtr2i 2643 . . . . . . . 8 ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) = (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × (ℂ ∖ {0})))
4589subid1d 10366 . . . . . . . . 9 (𝜑 → ((𝐹𝑋) − 0) = (𝐹𝑋))
459 txtopon 21375 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) → ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ∈ (TopOn‘(ℂ × ℂ)))
460433, 433, 459mp2an 707 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ∈ (TopOn‘(ℂ × ℂ))
461460toponunii 20702 . . . . . . . . . . . . 13 (ℂ × ℂ) = ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld))
462461restid 16075 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ∈ (TopOn‘(ℂ × ℂ)) → (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × ℂ)) = ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)))
463460, 462ax-mp 5 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × ℂ)) = ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld))
464463eqcomi 2629 . . . . . . . . . 10 ((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) = (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) ↾t (ℂ × ℂ))
465 limcresi 23630 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴) ⊆ (((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴)
466 ioossre 12220 . . . . . . . . . . . . . 14 (𝐴(,)𝑋) ⊆ ℝ
467 resmpt 5437 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)))
468466, 467ax-mp 5 . . . . . . . . . . . . 13 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋))
469468oveq1i 6645 . . . . . . . . . . . 12 (((𝑧 ∈ ℝ ↦ (𝐹𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴)
470465, 469sseqtri 3629 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴)
471 cncfmptc 22695 . . . . . . . . . . . . 13 (((𝐹𝑋) ∈ ℝ ∧ ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑧 ∈ ℝ ↦ (𝐹𝑋)) ∈ (ℝ–cn→ℝ))
4728, 144, 144, 471syl3anc 1324 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ ℝ ↦ (𝐹𝑋)) ∈ (ℝ–cn→ℝ))
473 eqidd 2621 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝐹𝑋) = (𝐹𝑋))
474472, 40, 473cnmptlimc 23635 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ ℝ ↦ (𝐹𝑋)) lim 𝐴))
475470, 474sseldi 3593 . . . . . . . . . 10 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑋)) lim 𝐴))
476 limcresi 23630 . . . . . . . . . . . 12 (𝐹 lim 𝐴) ⊆ ((𝐹 ↾ (𝐴(,)𝑋)) lim 𝐴)
4771, 107feqresmpt 6237 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)))
478477oveq1d 6650 . . . . . . . . . . . 12 (𝜑 → ((𝐹 ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
479476, 478syl5sseq 3645 . . . . . . . . . . 11 (𝜑 → (𝐹 lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
480 lhop1.f0 . . . . . . . . . . 11 (𝜑 → 0 ∈ (𝐹 lim 𝐴))
481479, 480sseldd 3596 . . . . . . . . . 10 (𝜑 → 0 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐹𝑧)) lim 𝐴))
48247subcn 22650 . . . . . . . . . . 11 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
483 0cn 10017 . . . . . . . . . . . 12 0 ∈ ℂ
484 opelxpi 5138 . . . . . . . . . . . 12 (((𝐹𝑋) ∈ ℂ ∧ 0 ∈ ℂ) → ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ))
4859, 483, 484sylancl 693 . . . . . . . . . . 11 (𝜑 → ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ))
486461cncnpi 21063 . . . . . . . . . . 11 (( − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) ∧ ⟨(𝐹𝑋), 0⟩ ∈ (ℂ × ℂ)) → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), 0⟩))
487482, 485, 486sylancr 694 . . . . . . . . . 10 (𝜑 → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), 0⟩))
488438, 441, 447, 447, 47, 464, 475, 481, 487limccnp2 23637 . . . . . . . . 9 (𝜑 → ((𝐹𝑋) − 0) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐹𝑋) − (𝐹𝑧))) lim 𝐴))
489458, 488eqeltrrd 2700 . . . . . . . 8 (𝜑 → (𝐹𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐹𝑋) − (𝐹𝑧))) lim 𝐴))
49012subid1d 10366 . . . . . . . . 9 (𝜑 → ((𝐺𝑋) − 0) = (𝐺𝑋))
491 limcresi 23630 . . . . . . . . . . . 12 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴) ⊆ (((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴)
492 resmpt 5437 . . . . . . . . . . . . . 14 ((𝐴(,)𝑋) ⊆ ℝ → ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)))
493466, 492ax-mp 5 . . . . . . . . . . . . 13 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋))
494493oveq1i 6645 . . . . . . . . . . . 12 (((𝑧 ∈ ℝ ↦ (𝐺𝑋)) ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴)
495491, 494sseqtri 3629 . . . . . . . . . . 11 ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴)
496 cncfmptc 22695 . . . . . . . . . . . . 13 (((𝐺𝑋) ∈ ℝ ∧ ℝ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑧 ∈ ℝ ↦ (𝐺𝑋)) ∈ (ℝ–cn→ℝ))
49711, 144, 144, 496syl3anc 1324 . . . . . . . . . . . 12 (𝜑 → (𝑧 ∈ ℝ ↦ (𝐺𝑋)) ∈ (ℝ–cn→ℝ))
498 eqidd 2621 . . . . . . . . . . . 12 (𝑧 = 𝐴 → (𝐺𝑋) = (𝐺𝑋))
499497, 40, 498cnmptlimc 23635 . . . . . . . . . . 11 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ ℝ ↦ (𝐺𝑋)) lim 𝐴))
500495, 499sseldi 3593 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑋)) lim 𝐴))
501 limcresi 23630 . . . . . . . . . . . 12 (𝐺 lim 𝐴) ⊆ ((𝐺 ↾ (𝐴(,)𝑋)) lim 𝐴)
50210, 107feqresmpt 6237 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ↾ (𝐴(,)𝑋)) = (𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)))
503502oveq1d 6650 . . . . . . . . . . . 12 (𝜑 → ((𝐺 ↾ (𝐴(,)𝑋)) lim 𝐴) = ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
504501, 503syl5sseq 3645 . . . . . . . . . . 11 (𝜑 → (𝐺 lim 𝐴) ⊆ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
505 lhop1.g0 . . . . . . . . . . 11 (𝜑 → 0 ∈ (𝐺 lim 𝐴))
506504, 505sseldd 3596 . . . . . . . . . 10 (𝜑 → 0 ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (𝐺𝑧)) lim 𝐴))
507 opelxpi 5138 . . . . . . . . . . . 12 (((𝐺𝑋) ∈ ℂ ∧ 0 ∈ ℂ) → ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ))
50812, 483, 507sylancl 693 . . . . . . . . . . 11 (𝜑 → ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ))
509461cncnpi 21063 . . . . . . . . . . 11 (( − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) ∧ ⟨(𝐺𝑋), 0⟩ ∈ (ℂ × ℂ)) → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐺𝑋), 0⟩))
510482, 508, 509sylancr 694 . . . . . . . . . 10 (𝜑 → − ∈ ((((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) CnP (TopOpen‘ℂfld))‘⟨(𝐺𝑋), 0⟩))
511120, 124, 447, 447, 47, 464, 500, 506, 510limccnp2 23637 . . . . . . . . 9 (𝜑 → ((𝐺𝑋) − 0) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐺𝑋) − (𝐺𝑧))) lim 𝐴))
512490, 511eqeltrrd 2700 . . . . . . . 8 (𝜑 → (𝐺𝑋) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ ((𝐺𝑋) − (𝐺𝑧))) lim 𝐴))
513 eqid 2620 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
51447, 513divcn 22652 . . . . . . . . 9 / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld))
515 eldifsn 4308 . . . . . . . . . . 11 ((𝐺𝑋) ∈ (ℂ ∖ {0}) ↔ ((𝐺𝑋) ∈ ℂ ∧ (𝐺𝑋) ≠ 0))
51612, 21, 515sylanbrc 697 . . . . . . . . . 10 (𝜑 → (𝐺𝑋) ∈ (ℂ ∖ {0}))
517 opelxpi 5138 . . . . . . . . . 10 (((𝐹𝑋) ∈ ℂ ∧ (𝐺𝑋) ∈ (ℂ ∖ {0})) → ⟨(𝐹𝑋), (𝐺𝑋)⟩ ∈ (ℂ × (ℂ ∖ {0})))
5189, 516, 517syl2anc 692 . . . . . . . . 9 (𝜑 → ⟨(𝐹𝑋), (𝐺𝑋)⟩ ∈ (ℂ × (ℂ ∖ {0})))
519 resttopon 20946 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0})))
520433, 448, 519mp2an 707 . . . . . . . . . . . 12 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) ∈ (TopOn‘(ℂ ∖ {0}))
521 txtopon 21375 . . . . . . . . . . . 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}))))
522433, 520, 521mp2an 707 . . . . . . . . . . 11 ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) ∈ (TopOn‘(ℂ × (ℂ ∖ {0})))
523522toponunii 20702 . . . . . . . . . 10 (ℂ × (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))
524523cncnpi 21063 . . . . . . . . 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)⟩))
525514, 518, 524sylancr 694 . . . . . . . 8 (𝜑 → / ∈ ((((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) CnP (TopOpen‘ℂfld))‘⟨(𝐹𝑋), (𝐺𝑋)⟩))
526442, 445, 447, 449, 47, 457, 489, 512, 525limccnp2 23637 . . . . . . 7 (𝜑 → ((𝐹𝑋) / (𝐺𝑋)) ∈ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) lim 𝐴))
527442, 443, 213divcld 10786 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝐴(,)𝑋)) → (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧))) ∈ ℂ)
528527, 388fmptd 6371 . . . . . . . 8 (𝜑 → (𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))):(𝐴(,)𝑋)⟶ℂ)
529466, 143sstri 3604 . . . . . . . . 9 (𝐴(,)𝑋) ⊆ ℂ
530529a1i 11 . . . . . . . 8 (𝜑 → (𝐴(,)𝑋) ⊆ ℂ)
531528, 530, 59, 47ellimc2 23622 . . . . . . 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)))))
532526, 531mpbid 222 . . . . . 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))))
533532simprd 479 . . . . 5 (𝜑 → ∀𝑢 ∈ (TopOpen‘ℂfld)(((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐴𝑣 ∧ ((𝑧 ∈ (𝐴(,)𝑋) ↦ (((𝐹𝑋) − (𝐹𝑧)) / ((𝐺𝑋) − (𝐺𝑧)))) “ (𝑣 ∩ ((𝐴(,)𝑋) ∖ {𝐴}))) ⊆ 𝑢)))
534 eleq2 2688 . . . . . . 7 (𝑢 = {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} → (((𝐹𝑋) / (𝐺𝑋)) ∈ 𝑢 ↔ ((𝐹𝑋) / (𝐺𝑋)) ∈ {𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸}))
535 sseq2 3619 . . . . . . . . 9 (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}))
536535anbi2d 739 . . . . . . . 8 (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})))
537536rexbidv 3048 . . . . . . 7 (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})))
538534, 537imbi12d 334 . . . . . 6 (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}))))
539538rspcv 3300 . . . . 5 ({𝑥 ∈ ℂ ∣ ¬ (abs‘(𝑥𝐶)) ≤ 𝐸} ∈ (TopOpen‘ℂfld) → (∀𝑢 ∈ (TopOpen‘ℂfld)(((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}))))
540437, 533, 539sylc 65 . . . 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})))
541419, 540sylbird 250 . . 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})))
542413, 541mt3d 140 . 2 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) ≤ 𝐸)
54329recnd 10053 . . . 4 (𝜑𝐸 ∈ ℂ)
544543mulid2d 10043 . . 3 (𝜑 → (1 · 𝐸) = 𝐸)
545 1red 10040 . . . 4 (𝜑 → 1 ∈ ℝ)
546 1lt2 11179 . . . . 5 1 < 2
547546a1i 11 . . . 4 (𝜑 → 1 < 2)
548545, 31, 28, 547ltmul1dd 11912 . . 3 (𝜑 → (1 · 𝐸) < (2 · 𝐸))
549544, 548eqbrtrrd 4668 . 2 (𝜑𝐸 < (2 · 𝐸))
55027, 29, 32, 542, 549lelttrd 10180 1 (𝜑 → (abs‘(((𝐹𝑋) / (𝐺𝑋)) − 𝐶)) < (2 · 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  ∃wrex 2910  {crab 2913  Vcvv 3195   ∖ cdif 3564   ∩ cin 3566   ⊆ wss 3567  ∅c0 3907  {csn 4168  ⟨cop 4174   class class class wbr 4644   ↦ cmpt 4720   × cxp 5102  dom cdm 5104  ran crn 5105   ↾ cres 5106   “ cima 5107   ∘ ccom 5108   Fn wfn 5871  ⟶wf 5872  ‘cfv 5876  (class class class)co 6635  ℂcc 9919  ℝcr 9920  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  ℝ*cxr 10058   < clt 10059   ≤ cle 10060   − cmin 10251   / cdiv 10669  2c2 11055  ℝ+crp 11817  (,)cioo 12160  [,]cicc 12163  abscabs 13955   ↾t crest 16062  TopOpenctopn 16063  topGenctg 16079  ∞Metcxmt 19712  ballcbl 19714  ℂfldccnfld 19727  TopOnctopon 20696  Clsdccld 20801  intcnt 20802   Cn ccn 21009   CnP ccnp 21010   ×t ctx 21344  –cn→ccncf 22660   limℂ climc 23607   D cdv 23608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-addf 10000  ax-mulf 10001 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-iin 4514  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-of 6882  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-fi 8302  df-sup 8333  df-inf 8334  df-oi 8400  df-card 8750  df-cda 8975  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-q 11774  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-ioo 12164  df-ico 12166  df-icc 12167  df-fz 12312  df-fzo 12450  df-seq 12785  df-exp 12844  df-hash 13101  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-abs 13957  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-sets 15845  df-ress 15846  df-plusg 15935  df-mulr 15936  df-starv 15937  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-unif 15946  df-hom 15947  df-cco 15948  df-rest 16064  df-topn 16065  df-0g 16083  df-gsum 16084  df-topgen 16085  df-pt 16086  df-prds 16089  df-xrs 16143  df-qtop 16148  df-imas 16149  df-xps 16151  df-mre 16227  df-mrc 16228  df-acs 16230  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317  df-mulg 17522  df-cntz 17731  df-cmn 18176  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-mopn 19723  df-fbas 19724  df-fg 19725  df-cnfld 19728  df-top 20680  df-topon 20697  df-topsp 20718  df-bases 20731  df-cld 20804  df-ntr 20805  df-cls 20806  df-nei 20883  df-lp 20921  df-perf 20922  df-cn 21012  df-cnp 21013  df-haus 21100  df-cmp 21171  df-tx 21346  df-hmeo 21539  df-fil 21631  df-fm 21723  df-flim 21724  df-flf 21725  df-xms 22106  df-ms 22107  df-tms 22108  df-cncf 22662  df-limc 23611  df-dv 23612 This theorem is referenced by:  lhop1  23758
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