P. 140 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-seq 13901*	Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as ℕ or ℕ0) whose value at an index is a function of its previous value and the value of an input sequence at that index. This definition is complicated, but fortunately it is not intended to be used directly. Instead, the only purpose of this definition is to provide us with an object that has the properties expressed by seq1 13913 and seqp1 13915. Typically, those are the main theorems that would be used in practice. The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series. seq𝑀( + , 𝐹) ⇝ 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 15451), by climdm 15453 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ⇝ ‘seq1( + , 𝐹)) (provided the sequence converges) and evaluates to 2 in this example. Internally, the rec function generates as its values a set of ordered pairs starting at ⟨𝑀, (𝐹‘𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain. This definition has its roots in a series of theorems from om2uz0i 13846 through om2uzf1oi 13852, originally proved by Raph Levien for use with df-exp 13961 and later generalized for arbitrary recursive sequences. Definition df-sum 15586 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.)
⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
Theorem	seqex 13902	Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ seq𝑀( + , 𝐹) ∈ V
Theorem	seqeq1 13903	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))
Theorem	seqeq2 13904	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Theorem	seqeq3 13905	Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Theorem	seqeq1d 13906	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))
Theorem	seqeq2d 13907	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))
Theorem	seqeq3d 13908	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))
Theorem	seqeq123d 13909	Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
⊢ (𝜑 → 𝑀 = 𝑁)    &   ⊢ (𝜑 → + = 𝑄)    &   ⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))
Theorem	nfseq 13910	Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝑀    &   ⊢ Ⅎ𝑥 +    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥seq𝑀( + , 𝐹)
Theorem	seqval 13911*	Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω)    ⇒   ⊢ seq𝑀( + , 𝐹) = ran 𝑅
Theorem	seqfn 13912	The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , 𝐹) Fn (ℤ≥‘𝑀))
Theorem	seq1 13913	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
Theorem	seq1i 13914	Value of the sequence builder function at its initial value. (Contributed by Mario Carneiro, 30-Apr-2014.)
⊢ 𝑀 ∈ ℤ    &   ⊢ (𝜑 → (𝐹‘𝑀) = 𝐴)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = 𝐴)
Theorem	seqp1 13915	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Sep-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))
Theorem	seqexw 13916	Weak version of seqex 13902 that holds without ax-rep 5215. A sequence builder exists when its binary operation input exists and its starting index is an integer. (Contributed by Rohan Ridenour, 14-Aug-2023.)
⊢ + ∈ V    &   ⊢ 𝑀 ∈ ℤ    ⇒   ⊢ seq𝑀( + , 𝐹) ∈ V
Theorem	seqp1d 13917	Value of the sequence builder function at a successor, deduction form. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by AV, 3-May-2024.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑍)    &   ⊢ 𝐾 = (𝑁 + 1)    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐾) = 𝐵)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐴 + 𝐵))
Theorem	seqm1 13918	Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹‘𝑁)))
Theorem	seqcl2 13919*	Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑀 + 1)...𝑁)) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝐶)
Theorem	seqf2 13920*	Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝐶)
Theorem	seqcl 13921*	Closure properties of the recursive sequence builder. (Contributed by Mario Carneiro, 2-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)
Theorem	seqf 13922*	Range of the recursive sequence builder (special case of seqf2 13920). (Contributed by Mario Carneiro, 24-Jun-2013.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶𝑆)
Theorem	seqfveq2 13923*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))
Theorem	seqfeq2 13924*	Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘(𝐾 + 1))) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝐾)) = seq𝐾( + , 𝐺))
Theorem	seqfveq 13925*	Equality of sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Theorem	seqfeq 13926*	Equality of sequences. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))
Theorem	seqshft2 13927*	Shifting the index set of a sequence. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))
Theorem	seqres 13928	Restricting its characteristic function to (ℤ≥‘𝑀) does not affect the seq function. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝐹 ↾ (ℤ≥‘𝑀))) = seq𝑀( + , 𝐹))
Theorem	serf 13929*	An infinite series of complex terms is a function from ℕ to ℂ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
Theorem	serfre 13930*	An infinite series of real numbers is a function from ℕ to ℝ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
Theorem	monoord 13931*	Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁))
Theorem	monoord2 13932*	Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀))
Theorem	sermono 13933*	The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	seqsplit 13934*	Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐾...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
Theorem	seq1p 13935*	Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))
Theorem	seqcaopr3 13936*	Lemma for seqcaopr2 13937. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1)))))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqcaopr2 13937*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqcaopr 13938*	The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqf1olem2a 13939*	Lemma for seqf1o 13942. (Contributed by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾)))
Theorem	seqf1olem1 13940*	Lemma for seqf1o 13942. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))    &   ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)    &   ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))    &   ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1))    ⇒   ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
Theorem	seqf1olem2 13941*	Lemma for seqf1o 13942. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)))    &   ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶)    &   ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))    &   ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1))    &   ⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1)))
Theorem	seqf1o 13942*	Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
Theorem	seradd 13943*	The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	sersub 13944*	The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqid3 13945*	A sequence that consists entirely of "zeroes" sums to "zero". More precisely, a constant sequence with value an element which is a + -idempotent sums (or "+'s") to that element. (Contributed by Mario Carneiro, 15-Dec-2014.)
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Theorem	seqid 13946*	Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (𝐹‘𝑁) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹))
Theorem	seqid2 13947*	The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)    &   ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))    &   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))
Theorem	seqhomo 13948*	Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))
Theorem	seqz 13949*	If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍)    &   ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍)    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)
Theorem	seqfeq4 13950*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁))
Theorem	seqfeq3 13951*	Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
Theorem	seqdistr 13952*	The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))
Theorem	ser0 13953	The value of the partial sums in a zero-valued infinite series. (Contributed by Mario Carneiro, 31-Aug-2013.) (Revised by Mario Carneiro, 15-Dec-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0)
Theorem	ser0f 13954	A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}))
Theorem	serge0 13955*	A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))
Theorem	serle 13956*	Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))
Theorem	ser1const 13957	Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴))
Theorem	seqof 13958*	Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))    ⇒   ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
Theorem	seqof2 13959*	Distribute function operation through a sequence. Maps-to notation version of seqof 13958. (Contributed by Mario Carneiro, 7-Jul-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (seq𝑀( ∘f + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁)))
5.6.7  Integer powers
Syntax	cexp 13960	Extend class notation to include exponentiation of a complex number to an integer power.
class ↑
Definition	df-exp 13961*	Define exponentiation of complex numbers with integer exponents. For example, (5↑2) = 25 (ex-exp 30420). Terminology: In general, "exponentiation" is the operation of raising a "base" 𝑥 to the power of the "exponent" 𝑦, resulting in the "power" (𝑥↑𝑦), also called "x to the power of y". In this case, "integer exponentiation" is the operation of raising a complex "base" 𝑥 to the power of an integer 𝑦, resulting in the "integer power" (𝑥↑𝑦). This definition is not meant to be used directly; instead, exp0 13964 and expp1 13967 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we do not have superscripts. 10-Jun-2005: The definition was extended from positive exponents to nonegative exponent, so that 0↑0 = 1, following standard convention, for instance Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 13965). 4-Jun-2014: The definition was extended to integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 30420). The case 𝑥 = 0, 𝑦 < 0 gives the "value" (1 / 0); relying on this should be avoided in applications. For a definition of exponentiation including complex exponents see df-cxp 26486 (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz 26596. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)
⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
Theorem	expval 13962	Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
Theorem	expnnval 13963	Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
Theorem	exp0 13964	Value of a complex number raised to the zeroth power. Under our definition, 0↑0 = 1 (0exp0e1 13965), following standard convention, for instance Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1)
Theorem	0exp0e1 13965	The zeroth power of zero equals one. See comment of exp0 13964. (Contributed by David A. Wheeler, 8-Dec-2018.)
⊢ (0↑0) = 1
Theorem	exp1 13966	Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴)
Theorem	expp1 13967	Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. When 𝐴 is nonzero, this holds for all integers 𝑁, see expneg 13968. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴))
Theorem	expneg 13968	Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
Theorem	expneg2 13969	Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))
Theorem	expn1 13970	A complex number raised to the negative one power is its reciprocal. When 𝐴 = 0, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ (𝐴 ∈ ℂ → (𝐴↑-1) = (1 / 𝐴))
Theorem	expcllem 13971*	Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
⊢ 𝐹 ⊆ ℂ    &   ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)    &   ⊢ 1 ∈ 𝐹    ⇒   ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ 𝐹)
Theorem	expcl2lem 13972*	Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ 𝐹 ⊆ ℂ    &   ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹)    &   ⊢ 1 ∈ 𝐹    &   ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐹)    ⇒   ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ 𝐹)
Theorem	nnexpcl 13973	Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ)
Theorem	nn0expcl 13974	Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0)
Theorem	zexpcl 13975	Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ)
Theorem	qexpcl 13976	Closure of exponentiation of rationals. For integer exponents, see qexpclz 13980. (Contributed by NM, 16-Dec-2005.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpcl 13977	Closure of exponentiation of reals. For integer exponents, see reexpclz 13981. (Contributed by NM, 14-Dec-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expcl 13978	Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 13983. (Contributed by NM, 26-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ)
Theorem	rpexpcl 13979	Closure law for integer exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+)
Theorem	qexpclz 13980	Closure of integer exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ)
Theorem	reexpclz 13981	Closure of integer exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
Theorem	expclzlem 13982	Lemma for expclz 13983. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0}))
Theorem	expclz 13983	Closure law for integer exponentiation of complex numnbers. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ)
Theorem	m1expcl2 13984	Closure of integer exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
Theorem	m1expcl 13985	Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
Theorem	zexpcld 13986	Closure of exponentiation of integers, deduction form. (Contributed by SN, 15-Sep-2024.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℤ)
Theorem	nn0expcli 13987	Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝑁 ∈ ℕ0    ⇒   ⊢ (𝐴↑𝑁) ∈ ℕ0
Theorem	nn0sqcl 13988	The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
⊢ (𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0)
Theorem	expm1t 13989	Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
Theorem	1exp 13990	Value of 1 raised to an integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
Theorem	expeq0 13991	A positive integer power is zero if and only if its base is zero. (Contributed by NM, 23-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
Theorem	expne0 13992	A positive integer power is nonzero if and only if its base is nonzero. (Contributed by NM, 6-May-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) ≠ 0 ↔ 𝐴 ≠ 0))
Theorem	expne0i 13993	An integer power is nonzero if its base is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0)
Theorem	expgt0 13994	A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 < 𝐴) → 0 < (𝐴↑𝑁))
Theorem	expnegz 13995	Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
Theorem	0exp 13996	Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0)
Theorem	expge0 13997	A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑁))
Theorem	expge1 13998	A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝐴) → 1 ≤ (𝐴↑𝑁))
Theorem	expgt1 13999	A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴) → 1 < (𝐴↑𝑁))
Theorem	mulexp 14000	Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁)))