HomeHome Intuitionistic Logic Explorer
Theorem List (p. 92 of 100)
< 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 - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiseqcl 9101* Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆)
 
Theoremiseqf 9102* Range of the recursive sequence builder. (Contributed by Jim Kingdon, 23-Jul-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥𝑍) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍𝑆)
 
Theoremiseqp1 9103* Value of the sequence builder function at a successor. (Contributed by Jim Kingdon, 31-May-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘(𝑁 + 1)) = ((seq𝑀( + , 𝐹, 𝑆)‘𝑁) + (𝐹‘(𝑁 + 1))))
 
Theoremiseqss 9104* Specifying a larger universe for seq. As long as 𝐹 and + are closed over 𝑆, then any set which contains 𝑆 can be used as the last argument to seq. This theorem does not allow 𝑇 to be a proper class, however. It also currently requires that + be closed over 𝑇 (as well as 𝑆). (Contributed by Jim Kingdon, 18-Aug-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑇𝑉)    &   (𝜑𝑆𝑇)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑇𝑦𝑇)) → (𝑥 + 𝑦) ∈ 𝑇)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐹, 𝑇))
 
Theoremiseqm1 9105* Value of the sequence builder function at a successor. (Contributed by Mario Carneiro, 24-Jun-2013.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = ((seq𝑀( + , 𝐹, 𝑆)‘(𝑁 − 1)) + (𝐹𝑁)))
 
Theoremiseqfveq2 9106* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺𝐾))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   (𝜑𝑁 ∈ (ℤ𝐾))    &   ((𝜑𝑘 ∈ ((𝐾 + 1)...𝑁)) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝐾( + , 𝐺, 𝑆)‘𝑁))
 
Theoremiseqfeq2 9107* Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
(𝜑𝐾 ∈ (ℤ𝑀))    &   (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝐾) = (𝐺𝐾))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ‘(𝐾 + 1))) → (𝐹𝑘) = (𝐺𝑘))       (𝜑 → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ𝐾)) = seq𝐾( + , 𝐺, 𝑆))
 
Theoremiseqfveq 9108* Equality of sequences. (Contributed by Jim Kingdon, 4-Jun-2020.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺𝑘))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq𝑀( + , 𝐺, 𝑆)‘𝑁))
 
Theoremiseqfeq 9109* Equality of sequences. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → (𝐹𝑘) = (𝐺𝑘))    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → seq𝑀( + , 𝐹, 𝑆) = seq𝑀( + , 𝐺, 𝑆))
 
Theoremiseqshft2 9110* Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐾 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝐾)))    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑𝑥 ∈ (ℤ‘(𝑀 + 𝐾))) → (𝐺𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺, 𝑆)‘(𝑁 + 𝐾)))
 
Theoremiserf 9111* An infinite series of complex terms is a function from to . (Contributed by Jim Kingdon, 15-Aug-2021.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)       (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
 
Theoremiserfre 9112* 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, ℝ):𝑍⟶ℝ)
 
Theoremmonoord 9113* 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 9114* 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 9115* 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 9116* 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)))    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ (ℤ𝐾))    &   ((𝜑𝑥 ∈ (ℤ𝐾)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝐾( + , 𝐹, 𝑆)‘𝑁) = ((seq𝐾( + , 𝐹, 𝑆)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
 
Theoremiseq1p 9117* 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)))    &   (𝜑𝑆𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = ((𝐹𝑀) + (seq(𝑀 + 1)( + , 𝐹, 𝑆)‘𝑁)))
 
Theoremiseqcaopr3 9118* 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 9119* 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 9120* 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)))
 
Theoremiseradd 9121* 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 9122* 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 9123* 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 9124* 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)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) = 𝑍)
 
Theoremiseqid 9125* 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)    &   (𝜑𝑆𝑉)    &   ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐹𝑥) ∈ 𝑆)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)       (𝜑 → (seq𝑀( + , 𝐹, 𝑆) ↾ (ℤ𝑁)) = seq𝑁( + , 𝐹, 𝑆))
 
Theoremiseqhomo 9126* 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))
 
Theoremiseqdistr 9127* 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 9128 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 9129 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}))
 
Theoremserige0 9130* 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 9131* 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))
 
3.5.9  Integer powers
 
Syntaxcexp 9132 Extend class notation to include exponentiation of a complex number to an integer power.
class
 
Definitiondf-iexp 9133* Define exponentiation to nonnegative integer powers. This definition is not meant to be used directly; instead, exp0 9137 and expp1 9140 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 9134 Lemma for expival 9135. 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 9135 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 9136 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 9137 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 9138 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 9139 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 9140 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 9141 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 9142 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 9143 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 9144* 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 9145* 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 9146 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ)
 
Theoremnn0expcl 9147 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℕ0)
 
Theoremzexpcl 9148 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℤ)
 
Theoremqexpcl 9149 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℚ)
 
Theoremreexpcl 9150 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℝ)
 
Theoremexpcl 9151 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ∈ ℂ)
 
Theoremrpexpcl 9152 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 9153 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℝ)
 
Theoremqexpclz 9154 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℚ)
 
Theoremm1expcl2 9155 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1})
 
Theoremm1expcl 9156 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ)
 
Theoremexpclzaplem 9157* Closure law for integer exponentiation. Lemma for expclzap 9158 and expap0i 9165. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0})
 
Theoremexpclzap 9158 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) ∈ ℂ)
 
Theoremnn0expcli 9159 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑁 ∈ ℕ0       (𝐴𝑁) ∈ ℕ0
 
Theoremnn0sqcl 9160 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0)
 
Theoremexpm1t 9161 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴))
 
Theorem1exp 9162 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 9163 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 9164 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 9164 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 9165 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 9166 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 9167 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 9168 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ∈ ℕ → (0↑𝑁) = 0)
 
Theoremexpge0 9169 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 9170 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 9171 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 9172 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 9173 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 9174 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 9175 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 9176 Lemma for expaddzap 9177. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴𝑀) · (𝐴𝑁)))
 
Theoremexpaddzap 9177 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 9178 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 9179 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))
 
Theoremexpsubap 9180 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 9181 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 9182 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 9183 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 9184 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 9185 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 9186 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 9187 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴𝐴𝐵)) → (𝐴𝑁) ≤ (𝐵𝑁))
 
Theoremexple1 9188 Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
(((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴𝐴 ≤ 1) ∧ 𝑁 ∈ ℕ0) → (𝐴𝑁) ≤ 1)
 
Theoremexpubnd 9189 An upper bound on 𝐴𝑁 when 2 ≤ 𝐴. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 2 ≤ 𝐴) → (𝐴𝑁) ≤ ((2↑𝑁) · ((𝐴 − 1)↑𝑁)))
 
Theoremsqval 9190 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
 
Theoremsqneg 9191 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
(𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
 
Theoremsqsubswap 9192 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝐵)↑2) = ((𝐵𝐴)↑2))
 
Theoremsqcl 9193 Closure of square. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ)
 
Theoremsqmul 9194 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2)))
 
Theoremsqeq0 9195 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0))
 
Theoremsqdivap 9196 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
 
Theoremsqne0 9197 A number is nonzero iff its square is nonzero. See also sqap0 9198 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0))
 
Theoremsqap0 9198 A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.)
(𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0))
 
Theoremresqcl 9199 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ)
 
Theoremsqgt0ap 9200 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐴 # 0) → 0 < (𝐴↑2))
    < 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 9901-9995
  Copyright terms: Public domain < Previous  Next >