P. 157 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rlimcn2 15601*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))    ⇒   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
Theorem	climcn1 15602*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐵 ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐹‘𝐴))
Theorem	climcn2 15603*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ⇝ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))
Theorem	addcn2 15604*	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 23267 and df-cncf 24920 are not yet available to us. See addcn 24906 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝐵 + 𝐶))) < 𝐴))
Theorem	subcn2 15605*	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 − 𝑣) − (𝐵 − 𝐶))) < 𝐴))
Theorem	mulcn2 15606*	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 · 𝑣) − (𝐵 · 𝐶))) < 𝐴))
Theorem	reccn2 15607*	The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)
⊢ 𝑇 = (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2))    ⇒   ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
Theorem	cn1lem 15608*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝐹:ℂ⟶ℂ    &   ⊢ ((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) ≤ (abs‘(𝑧 − 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥))
Theorem	abscn2 15609*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((abs‘𝑧) − (abs‘𝐴))) < 𝑥))
Theorem	cjcn2 15610*	The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((∗‘𝑧) − (∗‘𝐴))) < 𝑥))
Theorem	recn2 15611*	The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℜ‘𝑧) − (ℜ‘𝐴))) < 𝑥))
Theorem	imcn2 15612*	The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℑ‘𝑧) − (ℑ‘𝐴))) < 𝑥))
Theorem	climcn1lem 15613*	The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ 𝐻:ℂ⟶ℂ    &   ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐻‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴))
Theorem	climabs 15614*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (abs‘𝐴))
Theorem	climcj 15615*	Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (∗‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (∗‘𝐴))
Theorem	climre 15616*	Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℜ‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (ℜ‘𝐴))
Theorem	climim 15617*	Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℑ‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (ℑ‘𝐴))
Theorem	rlimmptrcl 15618*	Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
Theorem	rlimabs 15619*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (abs‘𝐵)) ⇝𝑟 (abs‘𝐶))
Theorem	rlimcj 15620*	Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (∗‘𝐵)) ⇝𝑟 (∗‘𝐶))
Theorem	rlimre 15621*	Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (ℜ‘𝐵)) ⇝𝑟 (ℜ‘𝐶))
Theorem	rlimim 15622*	Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (ℑ‘𝐵)) ⇝𝑟 (ℑ‘𝐶))
Theorem	o1of2 15623*	Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ) → 𝑀 ∈ ℝ)    &   ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥𝑅𝑦) ∈ ℂ)    &   ⊢ (((𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (((abs‘𝑥) ≤ 𝑚 ∧ (abs‘𝑦) ≤ 𝑛) → (abs‘(𝑥𝑅𝑦)) ≤ 𝑀))    ⇒   ⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f 𝑅𝐺) ∈ 𝑂(1))
Theorem	o1add 15624	The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f + 𝐺) ∈ 𝑂(1))
Theorem	o1mul 15625	The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f · 𝐺) ∈ 𝑂(1))
Theorem	o1sub 15626	The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.)
⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1)) → (𝐹 ∘f − 𝐺) ∈ 𝑂(1))
Theorem	rlimo1 15627	Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ 𝑂(1))
Theorem	rlimdmo1 15628	A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ (𝐹 ∈ dom ⇝𝑟 → 𝐹 ∈ 𝑂(1))
Theorem	o1rlimmul 15629	The product of an eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐺 ⇝𝑟 0) → (𝐹 ∘f · 𝐺) ⇝𝑟 0)
Theorem	o1const 15630*	A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
Theorem	lo1const 15631*	A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
Theorem	lo1mptrcl 15632*	Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
Theorem	o1mptrcl 15633*	Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)
Theorem	o1add2 15634*	The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝑂(1))
Theorem	o1mul2 15635*	The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∈ 𝑂(1))
Theorem	o1sub2 15636*	The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝑂(1))
Theorem	lo1add 15637*	The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ ≤𝑂(1))
Theorem	lo1mul 15638*	The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ∈ ≤𝑂(1))
Theorem	lo1mul2 15639*	The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ ≤𝑂(1))
Theorem	o1dif 15640*	If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)))
Theorem	lo1sub 15641*	The difference of an eventually upper bounded function and an eventually bounded function is eventually upper bounded. The "correct" sharp result here takes the second function to be eventually lower bounded instead of just bounded, but our notation for this is simply (𝑥 ∈ 𝐴 ↦ -𝐶) ∈ ≤𝑂(1), so it is just a special case of lo1add 15637. (Contributed by Mario Carneiro, 31-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ ≤𝑂(1))
Theorem	climadd 15642*	Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵))
Theorem	climmul 15643*	Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))
Theorem	climsub 15644*	Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 − 𝐵))
Theorem	climaddc1 15645*	Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶))
Theorem	climaddc2 15646*	Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴))
Theorem	climmulc2 15647*	Limit of a sequence multiplied by a constant 𝐶. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 · 𝐴))
Theorem	climsubc1 15648*	Limit of a constant 𝐶 subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − 𝐶))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐴 − 𝐶))
Theorem	climsubc2 15649*	Limit of a constant 𝐶 minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 − (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 − 𝐴))
Theorem	climle 15650*	Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	climsqz 15651*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
Theorem	climsqz2 15652*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
Theorem	rlimadd 15653*	Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ⇝𝑟 (𝐷 + 𝐸))
Theorem	rlimsub 15654*	Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ⇝𝑟 (𝐷 − 𝐸))
Theorem	rlimmul 15655*	Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · 𝐶)) ⇝𝑟 (𝐷 · 𝐸))
Theorem	rlimdiv 15656*	Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)    &   ⊢ (𝜑 → 𝐸 ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶)) ⇝𝑟 (𝐷 / 𝐸))
Theorem	rlimneg 15657*	Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ -𝐵) ⇝𝑟 -𝐶)
Theorem	rlimle 15658*	Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → 𝐷 ≤ 𝐸)
Theorem	rlimsqzlem 15659*	Lemma for rlimsqz 15660 and rlimsqz2 15661. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → (abs‘(𝐶 − 𝐸)) ≤ (abs‘(𝐵 − 𝐷)))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸)
Theorem	rlimsqz 15660*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)
⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐵 ≤ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
Theorem	rlimsqz2 15661*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.)
⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐷 ≤ 𝐶)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
Theorem	lo1le 15662*	Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → 𝐶 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))
Theorem	o1le 15663*	Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → (abs‘𝐶) ≤ (abs‘𝐵))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
Theorem	rlimno1 15664*	A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / 𝐵)) ⇝𝑟 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ¬ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
Theorem	clim2ser 15665*	The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    ⇒   ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ (𝐴 − (seq𝑀( + , 𝐹)‘𝑁)))
Theorem	clim2ser2 15666*	The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → seq(𝑁 + 1)( + , 𝐹) ⇝ 𝐴)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (𝐴 + (seq𝑀( + , 𝐹)‘𝑁)))
Theorem	iserex 15667*	An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ ))
Theorem	isermulc2 15668*	Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ (𝐶 · 𝐴))
Theorem	climlec2 15669*	Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	iserle 15670*	Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ (𝜑 → seq𝑀( + , 𝐺) ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	iserge0 15671*	The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	climub 15672*	The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝐴)
Theorem	climserle 15673*	The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ 𝐴)
Theorem	isershft 15674	Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ seq(𝑀 + 𝑁)( + , (𝐹 shift 𝑁)) ⇝ 𝐴))
Theorem	isercolllem1 15675*	Lemma for isercoll 15678. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → (𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆)))
Theorem	isercolllem2 15676*	Lemma for isercoll 15678. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
Theorem	isercolllem3 15677*	Lemma for isercoll 15678. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))
Theorem	isercoll 15678*	Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴))
Theorem	isercoll2 15679*	Generalize isercoll 15678 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑊 = (ℤ≥‘𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))
Theorem	climsup 15680*	A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ sup(ran 𝐹, ℝ, < ))
Theorem	climcau 15681*	A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)
Theorem	climbdd 15682*	A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥)
Theorem	caucvgrlem 15683*	Lemma for caurcvgr 15684. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝐴 ((lim sup‘𝐹) ∈ ℝ ∧ ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < (3 · 𝑅))))
Theorem	caurcvgr 15684*	A Cauchy sequence of real numbers converges to its limit supremum. The third hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by Mario Carneiro, 7-May-2016.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ⇝𝑟 (lim sup‘𝐹))
Theorem	caucvgrlem2 15685*	Lemma for caucvgr 15686. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    &   ⊢ 𝐻:ℂ⟶ℝ    &   ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑗) ∈ ℂ) → (abs‘((𝐻‘(𝐹‘𝑘)) − (𝐻‘(𝐹‘𝑗)))) ≤ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))))    ⇒   ⊢ (𝜑 → (𝑛 ∈ 𝐴 ↦ (𝐻‘(𝐹‘𝑛))) ⇝𝑟 ( ⇝𝑟 ‘(𝐻 ∘ 𝐹)))
Theorem	caucvgr 15686*	A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐴 (𝑗 ≤ 𝑘 → (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝𝑟 )
Theorem	caurcvg 15687*	A Cauchy sequence of real numbers converges to its limit supremum. The fourth hypothesis specifies that 𝐹 is a Cauchy sequence. (Contributed by NM, 4-Apr-2005.) (Revised by AV, 12-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((𝐹‘𝑘) − (𝐹‘𝑚))) < 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹))
Theorem	caurcvg2 15688*	A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	caucvg 15689*	A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )
Theorem	caucvgb 15690*	A function is convergent if and only if it is Cauchy. Theorem 12-5.3 of [Gleason] p. 180. (Contributed by Mario Carneiro, 15-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − (𝐹‘𝑗))) < 𝑥)))
Theorem	serf0 15691*	If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 0)
Theorem	iseraltlem1 15692*	Lemma for iseralt 15695. A decreasing sequence with limit zero consists of positive terms. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍) → 0 ≤ (𝐺‘𝑁))
Theorem	iseraltlem2 15693*	Lemma for iseralt 15695. The terms of an alternating series form a chain of inequalities in alternate terms, so that for example 𝑆(1) ≤ 𝑆(3) ≤ 𝑆(5) ≤ ... and ... ≤ 𝑆(4) ≤ 𝑆(2) ≤ 𝑆(0) (assuming 𝑀 = 0 so that these terms are defined). (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾)))) ≤ ((-1↑𝑁) · (seq𝑀( + , 𝐹)‘𝑁)))
Theorem	iseraltlem3 15694*	Lemma for iseralt 15695. From iseraltlem2 15693, we have (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) and (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1), and we also have (-1↑𝑛) · 𝑆(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) for each 𝑛 by the definition of the partial sum 𝑆, so combining the inequalities we get (-1↑𝑛) · 𝑆(𝑛) − 𝐺(𝑛 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) = (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − 𝐺(𝑛 + 2𝑘 + 1) ≤ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) ≤ (-1↑𝑛) · 𝑆(𝑛) ≤ (-1↑𝑛) · 𝑆(𝑛) + 𝐺(𝑛 + 1), so ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘 + 1) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘 + 1) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1) and ∣ (-1↑𝑛) · 𝑆(𝑛 + 2𝑘) − (-1↑𝑛) · 𝑆(𝑛) ∣ = ∣ 𝑆(𝑛 + 2𝑘) − 𝑆(𝑛) ∣ ≤ 𝐺(𝑛 + 1). Thus, both even and odd partial sums are Cauchy if 𝐺 converges to 0. (Contributed by Mario Carneiro, 6-Apr-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ ((𝜑 ∧ 𝑁 ∈ 𝑍 ∧ 𝐾 ∈ ℕ0) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑁 + (2 · 𝐾))) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1)) ∧ (abs‘((seq𝑀( + , 𝐹)‘((𝑁 + (2 · 𝐾)) + 1)) − (seq𝑀( + , 𝐹)‘𝑁))) ≤ (𝐺‘(𝑁 + 1))))
Theorem	iseralt 15695*	The alternating series test. If 𝐺(𝑘) is a decreasing sequence that converges to 0, then Σ𝑘 ∈ 𝑍(-1↑𝑘) · 𝐺(𝑘) is a convergent series. (Note that the first term is positive if 𝑀 is even, and negative if 𝑀 is odd. If the parity of your series does not match up with this, you will need to post-compose the series with multiplication by -1 using isermulc2 15668.) (Contributed by Mario Carneiro, 7-Apr-2015.) (Proof shortened by AV, 9-Jul-2022.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 1)) ≤ (𝐺‘𝑘))    &   ⊢ (𝜑 → 𝐺 ⇝ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((-1↑𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
5.10.3  Finite and infinite sums
Syntax	csu 15696	Extend class notation to include finite and infinite summations. (An underscore was added to the ASCII token in order to facilitate set.mm text searches, since "sum" is a commonly used word in comments.)
class Σ𝑘 ∈ 𝐴 𝐵
Definition	df-sum 15697*	Define the sum of a series with an index set of integers 𝐴. The variable 𝑘 is normally a free variable in 𝐵, i.e., 𝐵 can be thought of as 𝐵(𝑘). This definition is the result of a collection of discussions over the most general definition for a sum that does not need the index set to have a specified ordering. This definition is in two parts, one for finite sums and one for subsets of the upper integers. When summing over a subset of the upper integers, we extend the index set to the upper integers by adding zero outside the domain, and then sum the set in order, setting the result to the limit of the partial sums, if it exists. This means that conditionally convergent sums can be evaluated meaningfully. For finite sums, we are explicitly order-independent, by picking any bijection to a 1-based finite sequence and summing in the induced order. These two methods of summation produce the same result on their common region of definition (i.e., finite sets of integers) by summo 15727. Examples: Σ𝑘 ∈ {1, 2, 4}𝑘 means 1 + 2 + 4 = 7, and Σ𝑘 ∈ ℕ(1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 15895). (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
Theorem	sumex 15698	A sum is a set. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Σ𝑘 ∈ 𝐴 𝐵 ∈ V
Theorem	sumeq1 15699	Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
Theorem	nfsum1 15700	Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
⊢ Ⅎ𝑘𝐴    ⇒   ⊢ Ⅎ𝑘Σ𝑘 ∈ 𝐴 𝐵