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Theorem List for Intuitionistic Logic Explorer - 10001-10100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcseq 10001 Extend class notation with three-argument recursive sequence builder.
class seq𝑀( + , 𝐹)

Definitiondf-iseq 10002* Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as or 0) 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 iseq1 10021 and iseqp1 10028. 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, ℚ) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹, ℚ)‘1) = 1, (seq1( + , 𝐹, ℚ)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹, ℚ) transforms a sequence 𝐹 into an infinite series.

Internally, the frec function generates as its values a set of ordered pairs starting at 𝑀, (𝐹𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain. (Contributed by Jim Kingdon, 29-May-2020.) Use df-seq3 10003 or dfseq3-2 10004 instead. (New usage is discouraged.)

seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)

Definitiondf-seq3 10003 Define a three-argument version of seq. By theorems such as iseqsst 10031, it should be capable of doing pretty much everything that the four-argument version can, and may eventually replace the four-argument version entirely. (Contributed by Jim Kingdon, 3-Oct-2022.)
seq𝑀( + , 𝐹) = seq𝑀( + , 𝐹, V)

Theoremdfseq3-2 10004* Define a general-purpose operation that builds a recursive sequence (i.e., a function on an upper integer set such as or 0) 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 seqf 10026, seq3-1 10023 and seq3p1 10030. 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.

Internally, the frec function generates as its values a set of ordered pairs starting at 𝑀, (𝐹𝑀)⟩, with the first member of each pair incremented by one in each successive value. So, the range of frec is exactly the sequence we want, and we just extract the range and throw away the domain.

Eventually, this will be the definition of seq, replacing df-iseq 10002 and df-seq3 10003.

(Contributed by NM, 18-Apr-2005.) (Revised by Jim Kingdon, 4-Nov-2022.)

seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)

Theoremseqex 10005 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
seq𝑀( + , 𝐹) ∈ V

Theoremiseqeq3 10006 Equality theorem for the sequence builder operation.

New proofs should use seqeq3 10009 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

(𝐹 = 𝐺 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))

Theoremseqeq1 10007 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Theoremseqeq2 10008 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Theoremseqeq3 10009 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Theoremseqeq1d 10010 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹))

Theoremseqeq2d 10011 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹))

Theoremseqeq3d 10012 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝐴 = 𝐵)       (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵))

Theoremseqeq123d 10013 Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
(𝜑𝑀 = 𝑁)    &   (𝜑+ = 𝑄)    &   (𝜑𝐹 = 𝐺)       (𝜑 → seq𝑀( + , 𝐹) = seq𝑁(𝑄, 𝐺))

Theoremnfiseq 10014 Hypothesis builder for the sequence builder operation.

New proofs should use nfseq 10015 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

𝑥𝑀    &   𝑥 +    &   𝑥𝐹    &   𝑥𝑆       𝑥seq𝑀( + , 𝐹, 𝑆)

Theoremnfseq 10015 Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝑀    &   𝑥 +    &   𝑥𝐹       𝑥seq𝑀( + , 𝐹)

Theoremiseqovex 10016* Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       ((𝜑 ∧ (𝑥 ∈ (ℤ𝑀) ∧ 𝑦𝑆)) → (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆)

Theoremiseqval 10017* Value of the sequence builder function.

There should be no need for new usages of this theorem because once we have proved theorems seqf 10026, seq3-1 10023 and seq3p1 10030 future development can be done in terms of those.

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑆 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran 𝑅)

Theoremiseqvalcbv 10018* Changing the bound variables in an expression which appears in some seq related proofs. (Contributed by Jim Kingdon, 28-Apr-2022.)
frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩) = frec((𝑎 ∈ (ℤ𝑀), 𝑏𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ𝑀), 𝑑𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹𝑀)⟩)

Theoremiseqvalt 10019* Value of the sequence builder function.

There should be no need for new usages of this theorem because once we have proved theorems seqf 10026, seq3-1 10023 and seq3p1 10030 future development can be done in terms of those.

(Contributed by Jim Kingdon, 27-Apr-2022.) (New usage is discouraged.)

(𝜑𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)       (𝜑 → seq𝑀( + , 𝐹, 𝑇) = ran 𝑅)

Theoremseq3val 10020* Value of the sequence builder function. This helps expand the definition although there should be little need for it once we have proved seqf 10026, seq3-1 10023 and seq3p1 10030, as further development can be done in terms of those. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 4-Nov-2022.)
(𝜑𝑀 ∈ ℤ)    &   𝑅 = frec((𝑥 ∈ (ℤ𝑀), 𝑦 ∈ V ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ𝑀), 𝑤𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹𝑀)⟩)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹) = ran 𝑅)

Theoremiseq1 10021* Value of the sequence builder function at its initial value.

New proofs should use seq3-1 10023 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑀) = (𝐹𝑀))

Theoremiseq1t 10022* Value of the sequence builder function at its initial value.

New proofs should use seq3-1 10023 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)       (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘𝑀) = (𝐹𝑀))

Theoremseq3-1 10023* Value of the sequence builder function at its initial value. (Contributed by Jim Kingdon, 3-Oct-2022.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹𝑀))

Theoremiseqfcl 10024* Range of the recursive sequence builder. New proofs should use seqf 10026 instead (together with iseqsst 10031 or iseqseq3 10041 if need be). (Contributed by Jim Kingdon, 11-Apr-2022.) (New usage is discouraged.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍𝑆)

Theoremiseqfclt 10025* Range of the recursive sequence builder.

New proofs should use seqf 10026 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 26-Apr-2022.) (New usage is discouraged.)

𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)       (𝜑 → seq𝑀( + , 𝐹, 𝑇):𝑍𝑆)

Theoremseqf 10026* Range of the recursive sequence builder. (Contributed by Mario Carneiro, 24-Jun-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹):𝑍𝑆)

Theoremiseqcl 10027* Closure property of the recursive sequence builder.

New proofs should use seqf 10026 or seq3clss 10032 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 1-Jun-2020.) (New usage is discouraged.)

(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)

Theoremiseqp1 10028* Value of the sequence builder function at a successor.

New proofs should use seq3p1 10030 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 31-May-2020.) (New usage is discouraged.)

(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))

Theoremiseqp1t 10029* Value of the sequence builder function at a successor.

New proofs should use seq3p1 10030 instead (together with iseqsst 10031 or iseqseq3 10041 if need be).

(Contributed by Jim Kingdon, 30-Apr-2022.) (New usage is discouraged.)

(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)       (𝜑 → (seq𝑀( + , 𝐹, 𝑇)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑇)‘𝑁) + (𝐹‘(𝑁 + 1))))

Theoremseq3p1 10030* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 30-Apr-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 + 1))))

Theoremiseqsst 10031* Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any class which contains 𝑆 can be used as the last argument to seq.

Together with df-seq3 10003 it can be used to convert between the df-iseq 10002 syntax and the df-seq3 10003 syntax (in many cases iseqseq3 10041 is an even more convenient way to do this).

(Contributed by Jim Kingdon, 28-Apr-2022.)

(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑇)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))

Theoremseq3clss 10032* Closure property of the recursive sequence builder. (Contributed by Jim Kingdon, 28-Sep-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑇)    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑆𝑇)    &   ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ 𝑆)

Theoremseq3m1 10033* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Jim Kingdon, 3-Nov-2022.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((seq𝑀( + , 𝐹)‘(𝑁 − 1)) + (𝐹𝑁)))

Theoremseq3fveq2 10034* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝐾( + , 𝐺)‘𝑁))

Theoremseq3feq2 10035* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (𝐺𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ‘(𝐾 + 1))) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ𝐾)) = seq𝐾( + , 𝐺))

Theoremseq3fveq 10036* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Theoremseq3feq 10037* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺))

Theoremseq3shft2 10038* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾)))

Theoremserf 10039* An infinite series of complex terms is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)

Theoremserfre 10040* An infinite series of real numbers is a function from to . (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)       (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)

Theoremiseqseq3 10041* Equality of seq𝑀( + , 𝐹, ℂ) and seq𝑀( + , 𝐹). (Contributed by Jim Kingdon, 4-Oct-2022.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹, ℂ) = seq𝑀( + , 𝐹))

Theoremmonoord 10042* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))       (𝜑 → (𝐹𝑀) ≤ (𝐹𝑁))

Theoremmonoord2 10043* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹𝑘))       (𝜑 → (𝐹𝑁) ≤ (𝐹𝑀))

Theoremser3mono 10044* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹𝑥))       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁))

Theoremseq3split 10045* Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Theoremseq3-1p 10046* Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁)))

Theoremseq3caopr3 10047* Lemma for seq3caopr2 10048. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))    &   ((𝜑𝑛 ∈ (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)))

Theoremseq3caopr2 10048* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)    &   ((𝜑 ∧ ((𝑥𝑆𝑦𝑆) ∧ (𝑧𝑆𝑤𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘)𝑄(𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁)))

Theoremseq3caopr 10049* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))

Theoremiseqf1olemkle 10050* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)       (𝜑𝐾 ≤ (𝐽𝐾))

Theoremiseqf1olemklt 10051* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))       (𝜑𝐾 < (𝐽𝐾))

Theoremiseqf1olemqcl 10052 Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))       (𝜑 → if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)) ∈ (𝑀...𝑁))

Theoremiseqf1olemqval 10053* Lemma for seq3f1o 10070. Value of the function 𝑄. (Contributed by Jim Kingdon, 28-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑 → (𝑄𝐴) = if(𝐴 ∈ (𝐾...(𝐽𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽𝐴)))

Theoremiseqf1olemnab 10054* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑 → ¬ (𝐴 ∈ (𝐾...(𝐽𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(𝐽𝐾))))

Theoremiseqf1olemab 10055* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑𝐴 ∈ (𝐾...(𝐽𝐾)))    &   (𝜑𝐵 ∈ (𝐾...(𝐽𝐾)))       (𝜑𝐴 = 𝐵)

Theoremiseqf1olemnanb 10056* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑 → ¬ 𝐴 ∈ (𝐾...(𝐽𝐾)))    &   (𝜑 → ¬ 𝐵 ∈ (𝐾...(𝐽𝐾)))       (𝜑𝐴 = 𝐵)

Theoremiseqf1olemqf 10057* Lemma for seq3f1o 10070. Domain and codomain of 𝑄. (Contributed by Jim Kingdon, 26-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑𝑄:(𝑀...𝑁)⟶(𝑀...𝑁))

Theoremiseqf1olemmo 10058* Lemma for seq3f1o 10070. Showing that 𝑄 is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   (𝜑𝐵 ∈ (𝑀...𝑁))    &   (𝜑 → (𝑄𝐴) = (𝑄𝐵))       (𝜑𝐴 = 𝐵)

Theoremiseqf1olemqf1o 10059* Lemma for seq3f1o 10070. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))       (𝜑𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))

Theoremiseqf1olemqk 10060* Lemma for seq3f1o 10070. 𝑄 is constant for one more position than 𝐽 is. (Contributed by Jim Kingdon, 21-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)       (𝜑 → ∀𝑥 ∈ (𝑀...𝐾)(𝑄𝑥) = 𝑥)

Theoremiseqf1olemjpcl 10061* Lemma for seq3f1o 10070. A closure lemma involving 𝐽 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐽 / 𝑓𝑃𝑥) ∈ 𝑆)

Theoremiseqf1olemqpcl 10062* Lemma for seq3f1o 10070. A closure lemma involving 𝑄 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝑄 / 𝑓𝑃𝑥) ∈ 𝑆)

Theoremiseqf1olemfvp 10063* Lemma for seq3f1o 10070. (Contributed by Jim Kingdon, 30-Aug-2022.)
(𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝑇:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑𝐴 ∈ (𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (𝑇 / 𝑓𝑃𝐴) = (𝐺‘(𝑇𝐴)))

Theoremseq3f1olemqsumkj 10064* Lemma for seq3f1o 10070. 𝑄 gives the same sum as 𝐽 in the range (𝐾...(𝐽𝐾)). (Contributed by Jim Kingdon, 29-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝐾( + , 𝐽 / 𝑓𝑃)‘(𝐽𝐾)) = (seq𝐾( + , 𝑄 / 𝑓𝑃)‘(𝐽𝐾)))

Theoremseq3f1olemqsumk 10065* Lemma for seq3f1o 10070. 𝑄 gives the same sum as 𝐽 in the range (𝐾...𝑁). (Contributed by Jim Kingdon, 22-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝐾( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝐾( + , 𝑄 / 𝑓𝑃)‘𝑁))

Theoremseq3f1olemqsum 10066* Lemma for seq3f1o 10070. 𝑄 gives the same sum as 𝐽. (Contributed by Jim Kingdon, 21-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑𝐾 ≠ (𝐽𝐾))    &   𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝑄 / 𝑓𝑃)‘𝑁))

Theoremseq3f1olemstep 10067* Lemma for seq3f1o 10070. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)    &   (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

Theoremseq3f1olemp 10068* Lemma for seq3f1o 10070. Existence of a constant permutation of (𝑀...𝑁) which leads to the same sum as the permutation 𝐹 itself. (Contributed by Jim Kingdon, 18-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))    &   𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))       (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

Theoremseq3f1oleml 10069* Lemma for seq3f1o 10070. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))       (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Theoremseq3f1o 10070* Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐻𝑘) = (𝐺‘(𝐹𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Theoremser3add 10071* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁)))

Theoremser3sub 10072* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) − (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁)))

Theoremseq3id3 10073* 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.) (Revised by Jim Kingdon, 8-Apr-2023.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Theoremseq3id 10074* 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 Jim Kingdon, 8-Apr-2023.)
((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑥)    &   (𝜑𝑍𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑 → (𝐹𝑁) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹𝑥) = 𝑍)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹))

Theoremseq3id2 10075* 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 Jim Kingdon, 12-Nov-2022.)
((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)    &   (𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))

Theoremseq3homo 10076* Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻𝑥)𝑄(𝐻𝑦)))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐻‘(𝐹𝑥)) = (𝐺𝑥))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆)       (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁))

Theoremseq3z 10077* 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.) (Revised by Jim Kingdon, 23-Apr-2023.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑 → (𝐹𝐾) = 𝑍)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Theoremseqfeq3 10078* 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.)
(𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦))       (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

Theoremseq3distr 10079* The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = (𝐶𝑇(𝐺𝑥)))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁)))

Theoremser0 10080 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 = (ℤ𝑀)       (𝑁𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0)

Theoremser0f 10081 A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0}))

Theoremfser0const 10082* Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (𝑛𝑍 ↦ if(𝑛𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑍 × {0}))

Theoremser3ge0 10083* A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁))

Theoremser3le 10084* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁))

3.6.6  Integer powers

Syntaxcexp 10085 Extend class notation to include exponentiation of a complex number to an integer power.
class

Definitiondf-exp 10086* Define exponentiation to nonnegative integer powers. For example, (5↑2) = 25 (ex-exp 12366).

This definition is not meant to be used directly; instead, exp0 10090 and expp1 10093 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 don't have superscripts.

10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 10091).

4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 12366). The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (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)))))

Theoremexp3vallem 10087 Lemma for exp3val 10088. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 # 0)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (seq1( · , (ℕ × {𝐴}))‘𝑁) # 0)

Theoremexp3val 10088 Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Theoremexpnnval 10089 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁))

Theoremexp0 10090 Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1, following the convention used by Gleason. Part of 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)

Theorem0exp0e1 10091 0↑0 = 1 (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.)
(0↑0) = 1

Theoremexp1 10092 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)

Theoremexpp1 10093 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

Theoremexpnegap0 10094 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴𝑁)))

Theoremexpineg2 10095 Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0)) → (𝐴𝑁) = (1 / (𝐴↑-𝑁)))

Theoremexpn1ap0 10096 A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴↑-1) = (1 / 𝐴))

Theoremexpcllem 10097* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
𝐹 ⊆ ℂ    &   ((𝑥𝐹𝑦𝐹) → (𝑥 · 𝑦) ∈ 𝐹)    &   1 ∈ 𝐹       ((𝐴𝐹𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ 𝐹)

Theoremexpcl2lemap 10098* Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
𝐹 ⊆ ℂ    &   ((𝑥𝐹𝑦𝐹) → (𝑥 · 𝑦) ∈ 𝐹)    &   1 ∈ 𝐹    &   ((𝑥𝐹𝑥 # 0) → (1 / 𝑥) ∈ 𝐹)       ((𝐴𝐹𝐴 # 0 ∧ 𝐵 ∈ ℤ) → (𝐴𝐵) ∈ 𝐹)

Theoremnnexpcl 10099 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ)

Theoremnn0expcl 10100 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ0)

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