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Theorem limccoap 13007
 Description: Composition of two limits. This theorem is only usable in the case where 𝑥 # 𝑋 implies R(x) # 𝐶 so it is less general than might appear at first. (Contributed by Mario Carneiro, 29-Dec-2016.) (Revised by Jim Kingdon, 18-Dec-2023.)
Hypotheses
Ref Expression
limccoap.r ((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 ∈ {𝑤𝐵𝑤 # 𝐶})
limccoap.s ((𝜑𝑦 ∈ {𝑤𝐵𝑤 # 𝐶}) → 𝑆 ∈ ℂ)
limccoap.c (𝜑𝐶 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) lim 𝑋))
limccoap.d (𝜑𝐷 ∈ ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) lim 𝐶))
limcco.1 (𝑦 = 𝑅𝑆 = 𝑇)
Assertion
Ref Expression
limccoap (𝜑𝐷 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑇) lim 𝑋))
Distinct variable groups:   𝑤,𝐴,𝑥   𝑤,𝐵,𝑥,𝑦   𝑤,𝐶,𝑥,𝑦   𝑥,𝐷,𝑦   𝑤,𝑅,𝑦   𝑥,𝑆   𝑦,𝑇   𝑤,𝑋,𝑥   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑦)   𝐷(𝑤)   𝑅(𝑥)   𝑆(𝑦,𝑤)   𝑇(𝑥,𝑤)   𝑋(𝑦)

Proof of Theorem limccoap
Dummy variables 𝑑 𝑒 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccoap.d . . . 4 (𝜑𝐷 ∈ ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) lim 𝐶))
2 apsscn 8505 . . . . . 6 {𝑤𝐵𝑤 # 𝐶} ⊆ ℂ
32a1i 9 . . . . 5 (𝜑 → {𝑤𝐵𝑤 # 𝐶} ⊆ ℂ)
4 limcrcl 12987 . . . . . . 7 (𝐷 ∈ ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) lim 𝐶) → ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆):dom (𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
51, 4syl 14 . . . . . 6 (𝜑 → ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆):dom (𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆)⟶ℂ ∧ dom (𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) ⊆ ℂ ∧ 𝐶 ∈ ℂ))
65simp3d 996 . . . . 5 (𝜑𝐶 ∈ ℂ)
7 limccoap.s . . . . 5 ((𝜑𝑦 ∈ {𝑤𝐵𝑤 # 𝐶}) → 𝑆 ∈ ℂ)
83, 6, 7limcmpted 12992 . . . 4 (𝜑 → (𝐷 ∈ ((𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ↦ 𝑆) lim 𝐶) ↔ (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒))))
91, 8mpbid 146 . . 3 (𝜑 → (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)))
109simpld 111 . 2 (𝜑𝐷 ∈ ℂ)
119simprd 113 . . 3 (𝜑 → ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒))
12 breq2 3969 . . . . . . . . . 10 (𝑣 = 𝑑 → ((abs‘(𝑅𝐶)) < 𝑣 ↔ (abs‘(𝑅𝐶)) < 𝑑))
1312imbi2d 229 . . . . . . . . 9 (𝑣 = 𝑑 → (((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣) ↔ ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑)))
1413rexralbidv 2483 . . . . . . . 8 (𝑣 = 𝑑 → (∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣) ↔ ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑)))
15 limccoap.c . . . . . . . . . . 11 (𝜑𝐶 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) lim 𝑋))
16 apsscn 8505 . . . . . . . . . . . . 13 {𝑤𝐴𝑤 # 𝑋} ⊆ ℂ
1716a1i 9 . . . . . . . . . . . 12 (𝜑 → {𝑤𝐴𝑤 # 𝑋} ⊆ ℂ)
18 limcrcl 12987 . . . . . . . . . . . . . 14 (𝐶 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) lim 𝑋) → ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅):dom (𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) ⊆ ℂ ∧ 𝑋 ∈ ℂ))
1915, 18syl 14 . . . . . . . . . . . . 13 (𝜑 → ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅):dom (𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅)⟶ℂ ∧ dom (𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) ⊆ ℂ ∧ 𝑋 ∈ ℂ))
2019simp3d 996 . . . . . . . . . . . 12 (𝜑𝑋 ∈ ℂ)
21 limccoap.r . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 ∈ {𝑤𝐵𝑤 # 𝐶})
222, 21sseldi 3126 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 ∈ ℂ)
2317, 20, 22limcmpted 12992 . . . . . . . . . . 11 (𝜑 → (𝐶 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑅) lim 𝑋) ↔ (𝐶 ∈ ℂ ∧ ∀𝑣 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣))))
2415, 23mpbid 146 . . . . . . . . . 10 (𝜑 → (𝐶 ∈ ℂ ∧ ∀𝑣 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣)))
2524simprd 113 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣))
2625ad2antrr 480 . . . . . . . 8 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) → ∀𝑣 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑣))
27 simpr 109 . . . . . . . 8 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) → 𝑑 ∈ ℝ+)
2814, 26, 27rspcdva 2821 . . . . . . 7 (((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) → ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑))
2928adantr 274 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) → ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑))
30 simp-5l 533 . . . . . . . . . . . . 13 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝜑)
3130, 21sylancom 417 . . . . . . . . . . . 12 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 ∈ {𝑤𝐵𝑤 # 𝐶})
32 breq1 3968 . . . . . . . . . . . . 13 (𝑤 = 𝑅 → (𝑤 # 𝐶𝑅 # 𝐶))
3332elrab 2868 . . . . . . . . . . . 12 (𝑅 ∈ {𝑤𝐵𝑤 # 𝐶} ↔ (𝑅𝐵𝑅 # 𝐶))
3431, 33sylib 121 . . . . . . . . . . 11 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → (𝑅𝐵𝑅 # 𝐶))
3534simprd 113 . . . . . . . . . 10 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑅 # 𝐶)
36 breq1 3968 . . . . . . . . . . . . 13 (𝑦 = 𝑅 → (𝑦 # 𝐶𝑅 # 𝐶))
37 fvoveq1 5841 . . . . . . . . . . . . . 14 (𝑦 = 𝑅 → (abs‘(𝑦𝐶)) = (abs‘(𝑅𝐶)))
3837breq1d 3975 . . . . . . . . . . . . 13 (𝑦 = 𝑅 → ((abs‘(𝑦𝐶)) < 𝑑 ↔ (abs‘(𝑅𝐶)) < 𝑑))
3936, 38anbi12d 465 . . . . . . . . . . . 12 (𝑦 = 𝑅 → ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) ↔ (𝑅 # 𝐶 ∧ (abs‘(𝑅𝐶)) < 𝑑)))
40 limcco.1 . . . . . . . . . . . . . 14 (𝑦 = 𝑅𝑆 = 𝑇)
4140fvoveq1d 5840 . . . . . . . . . . . . 13 (𝑦 = 𝑅 → (abs‘(𝑆𝐷)) = (abs‘(𝑇𝐷)))
4241breq1d 3975 . . . . . . . . . . . 12 (𝑦 = 𝑅 → ((abs‘(𝑆𝐷)) < 𝑒 ↔ (abs‘(𝑇𝐷)) < 𝑒))
4339, 42imbi12d 233 . . . . . . . . . . 11 (𝑦 = 𝑅 → (((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒) ↔ ((𝑅 # 𝐶 ∧ (abs‘(𝑅𝐶)) < 𝑑) → (abs‘(𝑇𝐷)) < 𝑒)))
44 simpllr 524 . . . . . . . . . . 11 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒))
4543, 44, 31rspcdva 2821 . . . . . . . . . 10 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → ((𝑅 # 𝐶 ∧ (abs‘(𝑅𝐶)) < 𝑑) → (abs‘(𝑇𝐷)) < 𝑒))
4635, 45mpand 426 . . . . . . . . 9 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → ((abs‘(𝑅𝐶)) < 𝑑 → (abs‘(𝑇𝐷)) < 𝑒))
4746imim2d 54 . . . . . . . 8 ((((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) ∧ 𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → (((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑) → ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒)))
4847ralimdva 2524 . . . . . . 7 (((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) ∧ 𝑢 ∈ ℝ+) → (∀𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑) → ∀𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒)))
4948reximdva 2559 . . . . . 6 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) → (∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑅𝐶)) < 𝑑) → ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒)))
5029, 49mpd 13 . . . . 5 ((((𝜑𝑒 ∈ ℝ+) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒)) → ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒))
5150rexlimdva2 2577 . . . 4 ((𝜑𝑒 ∈ ℝ+) → (∃𝑑 ∈ ℝ+𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒) → ∃𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒)))
5251ralimdva 2524 . . 3 (𝜑 → (∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑦 ∈ {𝑤𝐵𝑤 # 𝐶} ((𝑦 # 𝐶 ∧ (abs‘(𝑦𝐶)) < 𝑑) → (abs‘(𝑆𝐷)) < 𝑒) → ∀𝑒 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒)))
5311, 52mpd 13 . 2 (𝜑 → ∀𝑒 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒))
5440eleq1d 2226 . . . 4 (𝑦 = 𝑅 → (𝑆 ∈ ℂ ↔ 𝑇 ∈ ℂ))
557ralrimiva 2530 . . . . 5 (𝜑 → ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶}𝑆 ∈ ℂ)
5655adantr 274 . . . 4 ((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → ∀𝑦 ∈ {𝑤𝐵𝑤 # 𝐶}𝑆 ∈ ℂ)
5754, 56, 21rspcdva 2821 . . 3 ((𝜑𝑥 ∈ {𝑤𝐴𝑤 # 𝑋}) → 𝑇 ∈ ℂ)
5817, 20, 57limcmpted 12992 . 2 (𝜑 → (𝐷 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑇) lim 𝑋) ↔ (𝐷 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑢 ∈ ℝ+𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ((𝑥 # 𝑋 ∧ (abs‘(𝑥𝑋)) < 𝑢) → (abs‘(𝑇𝐷)) < 𝑒))))
5910, 53, 58mpbir2and 929 1 (𝜑𝐷 ∈ ((𝑥 ∈ {𝑤𝐴𝑤 # 𝑋} ↦ 𝑇) lim 𝑋))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  {crab 2439   ⊆ wss 3102   class class class wbr 3965   ↦ cmpt 4025  dom cdm 4583  ⟶wf 5163  ‘cfv 5167  (class class class)co 5818  ℂcc 7713   < clt 7895   − cmin 8029   # cap 8439  ℝ+crp 9542  abscabs 10879   limℂ climc 12983 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-icn 7810  ax-addcl 7811  ax-mulcl 7813 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fo 5173  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-pm 6589  df-ap 8440  df-limced 12985 This theorem is referenced by:  dvcoapbr  13031
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