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Theorem List for Intuitionistic Logic Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmonoord 9801* 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 9802* 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))

Theoremisermono 9803* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ ℝ)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹𝑥))       (𝜑 → (seq𝑀( + , 𝐹, ℝ)‘𝐾) ≤ (seq𝑀( + , 𝐹, ℝ)‘𝑁))

Theoremiseqsplit 9804* Split a sequence into two sequences. (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 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

Theoremiseq1p 9805* 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, 𝑆)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))

Theoremiseqcaopr3 9806* Lemma for iseqcaopr2 . (Contributed by Jim Kingdon, 16-Aug-2021.)
((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, 𝑆)‘𝑛)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹, 𝑆)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺, 𝑆)‘𝑛) + (𝐺‘(𝑛 + 1)))))    &   (𝜑𝑆𝑉)       (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

Theoremiseqcaopr2 9807* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
((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)))    &   (𝜑𝑆𝑉)       (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁)𝑄(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

Theoremiseqcaopr 9808* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Jim Kingdon, 17-Aug-2021.)
((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)))    &   (𝜑𝑆𝑉)       (𝜑 → (seq𝑀( + , 𝐻, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

Theoremiseqf1olemkle 9809* Lemma for iseqf1o 9829. (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 9810* Lemma for iseqf1o 9829. (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 9811 Lemma for iseqf1o 9829. (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 9812* Lemma for iseqf1o 9829. 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 9813* Lemma for iseqf1o 9829. (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 9814* Lemma for iseqf1o 9829. (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 9815* Lemma for iseqf1o 9829. (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 9816* Lemma for iseqf1o 9829. 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 9817* Lemma for iseqf1o 9829. 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 9818* Lemma for iseqf1o 9829. 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 9819* Lemma for iseqf1o 9829. 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 9820* Lemma for iseqf1o 9829. 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 9821* Lemma for iseqf1o 9829. 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 9822* Lemma for iseqf1o 9829. (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)))

Theoremiseqf1olemqsumkj 9823* Lemma for iseqf1o 9829. 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, 𝑆)‘(𝐽𝐾)) = (seq𝐾( + , 𝑄 / 𝑓𝑃, 𝑆)‘(𝐽𝐾)))

Theoremiseqf1olemqsumk 9824* Lemma for iseqf1o 9829. 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, 𝑆)‘𝑁) = (seq𝐾( + , 𝑄 / 𝑓𝑃, 𝑆)‘𝑁))

Theoremiseqf1olemqsum 9825* Lemma for iseqf1o 9829. 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, 𝑆)‘𝑁) = (seq𝑀( + , 𝑄 / 𝑓𝑃, 𝑆)‘𝑁))

Theoremiseqf1olemstep 9826* Lemma for iseqf1o 9829. 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, 𝑆)‘𝑁) = (seq𝑀( + , 𝐿, 𝑆)‘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, 𝑆)‘𝑁) = (seq𝑀( + , 𝐿, 𝑆)‘𝑁)))

Theoremiseqf1olemp 9827* Lemma for iseqf1o 9829. 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, 𝑆)‘𝑁) = (seq𝑀( + , 𝐿, 𝑆)‘𝑁)))

Theoremiseqf1oleml 9828* Lemma for iseqf1o 9829. 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, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))

Theoremiseqf1o 9829* 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 ∧ (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, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))

Theoremiseradd 9830* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) + (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻, ℂ)‘𝑁) = ((seq𝑀( + , 𝐹, ℂ)‘𝑁) + (seq𝑀( + , 𝐺, ℂ)‘𝑁)))

Theoremisersub 9831* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐻𝑘) = ((𝐹𝑘) − (𝐺𝑘)))       (𝜑 → (seq𝑀( + , 𝐻, ℂ)‘𝑁) = ((seq𝑀( + , 𝐹, ℂ)‘𝑁) − (seq𝑀( + , 𝐺, ℂ)‘𝑁)))

Theoremiseqid3 9832* A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑉)       (𝜑 → (seq𝑀( + , 𝐹, {𝑍})‘𝑁) = 𝑍)

Theoremiseqid3s 9833* A sequence that consists of zeroes up to 𝑁 sums to zero at 𝑁. In this case by "zero" we mean whatever the identity 𝑍 is for the operation +). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑 → (𝑍 + 𝑍) = 𝑍)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (𝑀...𝑁)) → (𝐹𝑥) = 𝑍)    &   (𝜑𝑍𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)

Theoremiseqid 9834* Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity 𝑍 is for operation +). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
((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, 𝑆) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹, 𝑆))

Theoremiseqid2 9835* 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 Jim Kingdon, 5-Mar-2022.)
((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑥)    &   (𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) ∈ 𝑆)    &   ((𝜑𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑥) = 𝑍)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (seq𝑀( + , 𝐹, 𝑆)‘𝑁))

Theoremiseqhomo 9836* Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝑆𝑉)    &   (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, 𝑆)‘𝑁)) = (seq𝑀(𝑄, 𝐺, 𝑆)‘𝑁))

Theoremiseqz 9837* 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 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥𝑆) → (𝑍 + 𝑥) = 𝑍)    &   ((𝜑𝑥𝑆) → (𝑥 + 𝑍) = 𝑍)    &   (𝜑𝐾 ∈ (𝑀...𝑁))    &   (𝜑𝑁𝑉)    &   (𝜑 → (𝐹𝐾) = 𝑍)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)

Theoremiseqdistr 9838* The distributive property for series. (Contributed by Jim Kingdon, 21-Aug-2021.)
((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦)))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) = (𝐶𝑇(𝐺𝑥)))    &   (𝜑𝑆𝑉)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺, 𝑆)‘𝑁)))

Theoremiser0 9839 The value of the partial sums in a zero-valued infinite series. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ𝑀)       (𝑁𝑍 → (seq𝑀( + , (𝑍 × {0}), ℂ)‘𝑁) = 0)

Theoremiser0f 9840 A zero-valued infinite series is equal to the constant zero function. (Contributed by Jim Kingdon, 19-Aug-2021.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0}), ℂ) = (𝑍 × {0}))

Theoremfser0const 9841* 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}))

Theoremserige0 9842* A finite sum of nonnegative terms is nonnegative. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → 0 ≤ (𝐹𝑘))       (𝜑 → 0 ≤ (seq𝑀( + , 𝐹, ℂ)‘𝑁))

Theoremserile 9843* Comparison of partial sums of two infinite series of reals. (Contributed by Jim Kingdon, 22-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐺𝑘) ∈ ℝ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) ≤ (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹, ℂ)‘𝑁) ≤ (seq𝑀( + , 𝐺, ℂ)‘𝑁))

3.6.6  Integer powers

Syntaxcexp 9844 Extend class notation to include exponentiation of a complex number to an integer power.
class

Definitiondf-iexp 9845* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9849 and expp1 9852 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. 4-Jun-2014: The definition was extended to include negative integer exponents. The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Jim Kingdon, 7-Jun-2020.)
↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}), ℂ)‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}), ℂ)‘-𝑦)))))

Theoremexpivallem 9846 Lemma for expival 9847. 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.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ) → (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁) # 0)

Theoremexpival 9847 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}), ℂ)‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}), ℂ)‘-𝑁)))))

Theoremexpinnval 9848 Value of exponentiation to positive integer powers. (Contributed by Jim Kingdon, 8-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = (seq1( · , (ℕ × {𝐴}), ℂ)‘𝑁))

Theoremexp0 9849 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 9850 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 9851 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 9852 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 9853 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 9854 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 9855 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 9856* 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 9857* 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 9858 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ)

Theoremnn0expcl 9859 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ0)

Theoremzexpcl 9860 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℤ)

Theoremqexpcl 9861 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℚ)

Theoremreexpcl 9862 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℝ)

Theoremexpcl 9863 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℂ)

Theoremrpexpcl 9864 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ+)

Theoremreexpclzap 9865 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ)

Theoremqexpclz 9866 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℚ)

Theoremm1expcl2 9867 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})

Theoremm1expcl 9868 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)

Theoremexpclzaplem 9869* Closure law for integer exponentiation. Lemma for expclzap 9870 and expap0i 9877. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})

Theoremexpclzap 9870 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℂ)

Theoremnn0expcli 9871 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝐴𝑁) ∈ ℕ0

Theoremnn0sqcl 9872 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0)

Theoremexpm1t 9873 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))

Theorem1exp 9874 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝑁 ∈ ℤ → (1↑𝑁) = 1)

Theoremexpap0 9875 Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 9876 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) # 0 ↔ 𝐴 # 0))

Theoremexpeq0 9876 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴𝑁) = 0 ↔ 𝐴 = 0))

Theoremexpap0i 9877 Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) # 0)

Theoremexpgt0 9878 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 < 𝐴) → 0 < (𝐴𝑁))

Theoremexpnegzap 9879 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴𝑁)))

Theorem0exp 9880 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ∈ ℕ → (0↑𝑁) = 0)

Theoremexpge0 9881 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴𝑁))

Theoremexpge1 9882 Nonnegative integer exponentiation with a mantissa greater than or equal to 1 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 ≤ (𝐴𝑁))

Theoremexpgt1 9883 Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴) → 1 < (𝐴𝑁))

Theoremmulexp 9884 Positive 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) = ((𝐴𝑁) · (𝐵𝑁)))

Theoremmulexpzap 9885 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴𝑁) · (𝐵𝑁)))

Theoremexprecap 9886 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴𝑁)))

Theoremexpadd 9887 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpaddzaplem 9888 Lemma for expaddzap 9889. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpaddzap 9889 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))

Theoremexpmul 9890 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴𝑀)↑𝑁))

Theoremexpmulzap 9891 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴𝑀)↑𝑁))

Theoremm1expeven 9892 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
(𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1)

Theoremexpsubap 9893 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀𝑁)) = ((𝐴𝑀) / (𝐴𝑁)))

Theoremexpp1zap 9894 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴𝑁) · 𝐴))

Theoremexpm1ap 9895 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴𝑁) / 𝐴))

Theoremexpdivap 9896 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴𝑁) / (𝐵𝑁)))

Theoremltexp2a 9897 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 < 𝐴𝑀 < 𝑁)) → (𝐴𝑀) < (𝐴𝑁))

Theoremleexp2a 9898 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴𝑁 ∈ (ℤ𝑀)) → (𝐴𝑀) ≤ (𝐴𝑁))

Theoremleexp2r 9899 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
(((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ (0 ≤ 𝐴𝐴 ≤ 1)) → (𝐴𝑁) ≤ (𝐴𝑀))

Theoremleexp1a 9900 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))

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