HomeHome Intuitionistic Logic Explorer
Theorem List (p. 99 of 114)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11354
  Copyright terms: Public domain < Previous  Next >