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Theorem List for Metamath Proof Explorer - 14001-14100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremserf 14001* An infinite series of complex terms is a function from β„• to β„‚. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„‚)
 
Theoremserfre 14002* An infinite series of real numbers is a function from β„• to ℝ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹):π‘βŸΆβ„)
 
Theoremmonoord 14003* Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘˜) ≀ (πΉβ€˜(π‘˜ + 1)))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘€) ≀ (πΉβ€˜π‘))
 
Theoremmonoord2 14004* Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜(π‘˜ + 1)) ≀ (πΉβ€˜π‘˜))    β‡’   (πœ‘ β†’ (πΉβ€˜π‘) ≀ (πΉβ€˜π‘€))
 
Theoremsermono 14005* The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-Jun-2013.)
(πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜πΎ))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ ℝ)    &   ((πœ‘ ∧ π‘₯ ∈ ((𝐾 + 1)...𝑁)) β†’ 0 ≀ (πΉβ€˜π‘₯))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜πΎ) ≀ (seq𝑀( + , 𝐹)β€˜π‘))
 
Theoremseqsplit 14006* Split a sequence into two sequences. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑀 + 1)))    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜πΎ))    &   ((πœ‘ ∧ π‘₯ ∈ (𝐾...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    β‡’   (πœ‘ β†’ (seq𝐾( + , 𝐹)β€˜π‘) = ((seq𝐾( + , 𝐹)β€˜π‘€) + (seq(𝑀 + 1)( + , 𝐹)β€˜π‘)))
 
Theoremseq1p 14007* Removing the first term from a sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜(𝑀 + 1)))    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = ((πΉβ€˜π‘€) + (seq(𝑀 + 1)( + , 𝐹)β€˜π‘)))
 
Theoremseqcaopr3 14008* Lemma for seqcaopr2 14009. (Contributed by Mario Carneiro, 25-Apr-2016.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯𝑄𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜)𝑄(πΊβ€˜π‘˜)))    &   ((πœ‘ ∧ 𝑛 ∈ (𝑀..^𝑁)) β†’ (((seq𝑀( + , 𝐹)β€˜π‘›)𝑄(seq𝑀( + , 𝐺)β€˜π‘›)) + ((πΉβ€˜(𝑛 + 1))𝑄(πΊβ€˜(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)β€˜π‘›) + (πΉβ€˜(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)β€˜π‘›) + (πΊβ€˜(𝑛 + 1)))))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘)𝑄(seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremseqcaopr2 14009* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯𝑄𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ ((π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑀 ∈ 𝑆))) β†’ ((π‘₯𝑄𝑧) + (𝑦𝑄𝑀)) = ((π‘₯ + 𝑦)𝑄(𝑧 + 𝑀)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜)𝑄(πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘)𝑄(seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremseqcaopr 14010* The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 30-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ 𝑆)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) + (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘) + (seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremseqf1olem2a 14011* Lemma for seqf1o 14014. (Contributed by Mario Carneiro, 24-Apr-2016.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐢 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐺:𝐴⟢𝐢)    &   (πœ‘ β†’ 𝐾 ∈ 𝐴)    &   (πœ‘ β†’ (𝑀...𝑁) βŠ† 𝐴)    β‡’   (πœ‘ β†’ ((πΊβ€˜πΎ) + (seq𝑀( + , 𝐺)β€˜π‘)) = ((seq𝑀( + , 𝐺)β€˜π‘) + (πΊβ€˜πΎ)))
 
Theoremseqf1olem1 14012* Lemma for seqf1o 14014. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐢 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:(𝑀...(𝑁 + 1))–1-1-ontoβ†’(𝑀...(𝑁 + 1)))    &   (πœ‘ β†’ 𝐺:(𝑀...(𝑁 + 1))⟢𝐢)    &   π½ = (π‘˜ ∈ (𝑀...𝑁) ↦ (πΉβ€˜if(π‘˜ < 𝐾, π‘˜, (π‘˜ + 1))))    &   πΎ = (β—‘πΉβ€˜(𝑁 + 1))    β‡’   (πœ‘ β†’ 𝐽:(𝑀...𝑁)–1-1-ontoβ†’(𝑀...𝑁))
 
Theoremseqf1olem2 14013* Lemma for seqf1o 14014. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐢 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:(𝑀...(𝑁 + 1))–1-1-ontoβ†’(𝑀...(𝑁 + 1)))    &   (πœ‘ β†’ 𝐺:(𝑀...(𝑁 + 1))⟢𝐢)    &   π½ = (π‘˜ ∈ (𝑀...𝑁) ↦ (πΉβ€˜if(π‘˜ < 𝐾, π‘˜, (π‘˜ + 1))))    &   πΎ = (β—‘πΉβ€˜(𝑁 + 1))    &   (πœ‘ β†’ βˆ€π‘”βˆ€π‘“((𝑓:(𝑀...𝑁)–1-1-ontoβ†’(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟢𝐢) β†’ (seq𝑀( + , (𝑔 ∘ 𝑓))β€˜π‘) = (seq𝑀( + , 𝑔)β€˜π‘)))    β‡’   (πœ‘ β†’ (seq𝑀( + , (𝐺 ∘ 𝐹))β€˜(𝑁 + 1)) = (seq𝑀( + , 𝐺)β€˜(𝑁 + 1)))
 
Theoremseqf1o 14014* Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐢 ∧ 𝑦 ∈ 𝐢)) β†’ (π‘₯ + 𝑦) = (𝑦 + π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) β†’ ((π‘₯ + 𝑦) + 𝑧) = (π‘₯ + (𝑦 + 𝑧)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐢 βŠ† 𝑆)    &   (πœ‘ β†’ 𝐹:(𝑀...𝑁)–1-1-ontoβ†’(𝑀...𝑁))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘₯) ∈ 𝐢)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = (πΊβ€˜(πΉβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = (seq𝑀( + , 𝐺)β€˜π‘))
 
Theoremseradd 14015* The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) + (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘) + (seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremsersub 14016* The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (π»β€˜π‘˜) = ((πΉβ€˜π‘˜) βˆ’ (πΊβ€˜π‘˜)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐻)β€˜π‘) = ((seq𝑀( + , 𝐹)β€˜π‘) βˆ’ (seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremseqid3 14017* 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.)
(πœ‘ β†’ (𝑍 + 𝑍) = 𝑍)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) = 𝑍)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = 𝑍)
 
Theoremseqid 14018* Discarding the first few terms of a sequence that starts with all zeroes (or any element which is a left-identity for +) has no effect on its sum. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (𝑍 + π‘₯) = π‘₯)    &   (πœ‘ β†’ 𝑍 ∈ 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ (πΉβ€˜π‘) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...(𝑁 βˆ’ 1))) β†’ (πΉβ€˜π‘₯) = 𝑍)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹) β†Ύ (β„€β‰₯β€˜π‘)) = seq𝑁( + , 𝐹))
 
Theoremseqid2 14019* The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯ + 𝑍) = π‘₯)    &   (πœ‘ β†’ 𝐾 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜πΎ))    &   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜πΎ) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ ((𝐾 + 1)...𝑁)) β†’ (πΉβ€˜π‘₯) = 𝑍)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜πΎ) = (seq𝑀( + , 𝐹)β€˜π‘))
 
Theoremseqhomo 14020* Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π»β€˜(π‘₯ + 𝑦)) = ((π»β€˜π‘₯)𝑄(π»β€˜π‘¦)))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (π»β€˜(πΉβ€˜π‘₯)) = (πΊβ€˜π‘₯))    β‡’   (πœ‘ β†’ (π»β€˜(seq𝑀( + , 𝐹)β€˜π‘)) = (seq𝑀(𝑄, 𝐺)β€˜π‘))
 
Theoremseqz 14021* 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.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (𝑍 + π‘₯) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑆) β†’ (π‘₯ + 𝑍) = 𝑍)    &   (πœ‘ β†’ 𝐾 ∈ (𝑀...𝑁))    &   (πœ‘ β†’ 𝑁 ∈ 𝑉)    &   (πœ‘ β†’ (πΉβ€˜πΎ) = 𝑍)    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = 𝑍)
 
Theoremseqfeq4 14022* Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) = (π‘₯𝑄𝑦))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = (seq𝑀(𝑄, 𝐹)β€˜π‘))
 
Theoremseqfeq3 14023* 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘₯ ∈ (β„€β‰₯β€˜π‘€)) β†’ (πΉβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) = (π‘₯𝑄𝑦))    β‡’   (πœ‘ β†’ seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))
 
Theoremseqdistr 14024* The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ + 𝑦) ∈ 𝑆)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (𝐢𝑇(π‘₯ + 𝑦)) = ((𝐢𝑇π‘₯) + (𝐢𝑇𝑦)))    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘₯) ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) = (𝐢𝑇(πΊβ€˜π‘₯)))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) = (𝐢𝑇(seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremser0 14025 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 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑁 ∈ 𝑍 β†’ (seq𝑀( + , (𝑍 Γ— {0}))β€˜π‘) = 0)
 
Theoremser0f 14026 A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.)
𝑍 = (β„€β‰₯β€˜π‘€)    β‡’   (𝑀 ∈ β„€ β†’ seq𝑀( + , (𝑍 Γ— {0})) = (𝑍 Γ— {0}))
 
Theoremserge0 14027* A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ 0 ≀ (πΉβ€˜π‘˜))    β‡’   (πœ‘ β†’ 0 ≀ (seq𝑀( + , 𝐹)β€˜π‘))
 
Theoremserle 14028* Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΊβ€˜π‘˜) ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘˜) ≀ (πΊβ€˜π‘˜))    β‡’   (πœ‘ β†’ (seq𝑀( + , 𝐹)β€˜π‘) ≀ (seq𝑀( + , 𝐺)β€˜π‘))
 
Theoremser1const 14029 Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ (seq1( + , (β„• Γ— {𝐴}))β€˜π‘) = (𝑁 Β· 𝐴))
 
Theoremseqof 14030* Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘₯ ∈ (𝑀...𝑁)) β†’ (πΉβ€˜π‘₯) = (𝑧 ∈ 𝐴 ↦ (πΊβ€˜π‘₯)))    β‡’   (πœ‘ β†’ (seq𝑀( ∘f + , 𝐹)β€˜π‘) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)β€˜π‘)))
 
Theoremseqof2 14031* Distribute function operation through a sequence. Maps-to notation version of seqof 14030. (Contributed by Mario Carneiro, 7-Jul-2017.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ (𝑀...𝑁) βŠ† 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑧 ∈ 𝐴)) β†’ 𝑋 ∈ π‘Š)    β‡’   (πœ‘ β†’ (seq𝑀( ∘f + , (π‘₯ ∈ 𝐡 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))β€˜π‘) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (π‘₯ ∈ 𝐡 ↦ 𝑋))β€˜π‘)))
 
5.6.7  Integer powers
 
Syntaxcexp 14032 Extend class notation to include exponentiation of a complex number to an integer power.
class ↑
 
Definitiondf-exp 14033* Define exponentiation of complex numbers with integer exponents. For example, (5↑2) = 25 (ex-exp 29971). Terminology: In general, "exponentiation" is the operation of raising a "base" π‘₯ to the power of the "exponent" 𝑦, resulting in the "power" (π‘₯↑𝑦), also called "x to the power of y". In this case, "integer exponentiation" is the operation of raising a complex "base" π‘₯ to the power of an integer 𝑦, resulting in the "integer power" (π‘₯↑𝑦).

This definition is not meant to be used directly; instead, exp0 14036 and expp1 14039 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we do not have superscripts.

10-Jun-2005: The definition was extended from positive exponents to nonegative exponent, so that 0↑0 = 1, following standard convention, for instance Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 14037).

4-Jun-2014: The definition was extended to integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 29971). The case π‘₯ = 0, 𝑦 < 0 gives the "value" (1 / 0); relying on this should be avoided in applications.

For a definition of exponentiation including complex exponents see df-cxp 26303 (complex exponentiation). Both definitions are equivalent for integer exponents, see cxpexpz 26412. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.)

↑ = (π‘₯ ∈ β„‚, 𝑦 ∈ β„€ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( Β· , (β„• Γ— {π‘₯}))β€˜π‘¦), (1 / (seq1( Β· , (β„• Γ— {π‘₯}))β€˜-𝑦)))))
 
Theoremexpval 14034 Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( Β· , (β„• Γ— {𝐴}))β€˜π‘), (1 / (seq1( Β· , (β„• Γ— {𝐴}))β€˜-𝑁)))))
 
Theoremexpnnval 14035 Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ (𝐴↑𝑁) = (seq1( Β· , (β„• Γ— {𝐴}))β€˜π‘))
 
Theoremexp0 14036 Value of a complex number raised to the zeroth power. Under our definition, 0↑0 = 1 (0exp0e1 14037), following standard convention, for instance Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝐴 ∈ β„‚ β†’ (𝐴↑0) = 1)
 
Theorem0exp0e1 14037 The zeroth power of zero equals one. See comment of exp0 14036. (Contributed by David A. Wheeler, 8-Dec-2018.)
(0↑0) = 1
 
Theoremexp1 14038 Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.)
(𝐴 ∈ β„‚ β†’ (𝐴↑1) = 𝐴)
 
Theoremexpp1 14039 Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. When 𝐴 is nonzero, this holds for all integers 𝑁, see expneg 14040. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) Β· 𝐴))
 
Theoremexpneg 14040 Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
 
Theoremexpneg2 14041 Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„‚ ∧ -𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) = (1 / (𝐴↑-𝑁)))
 
Theoremexpn1 14042 A complex number raised to the negative one power is its reciprocal. When 𝐴 = 0, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.)
(𝐴 ∈ β„‚ β†’ (𝐴↑-1) = (1 / 𝐴))
 
Theoremexpcllem 14043* Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.)
𝐹 βŠ† β„‚    &   ((π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) β†’ (π‘₯ Β· 𝑦) ∈ 𝐹)    &   1 ∈ 𝐹    β‡’   ((𝐴 ∈ 𝐹 ∧ 𝐡 ∈ β„•0) β†’ (𝐴↑𝐡) ∈ 𝐹)
 
Theoremexpcl2lem 14044* Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
𝐹 βŠ† β„‚    &   ((π‘₯ ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) β†’ (π‘₯ Β· 𝑦) ∈ 𝐹)    &   1 ∈ 𝐹    &   ((π‘₯ ∈ 𝐹 ∧ π‘₯ β‰  0) β†’ (1 / π‘₯) ∈ 𝐹)    β‡’   ((𝐴 ∈ 𝐹 ∧ 𝐴 β‰  0 ∧ 𝐡 ∈ β„€) β†’ (𝐴↑𝐡) ∈ 𝐹)
 
Theoremnnexpcl 14045 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ β„• ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ β„•)
 
Theoremnn0expcl 14046 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ β„•0)
 
Theoremzexpcl 14047 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ β„€)
 
Theoremqexpcl 14048 Closure of exponentiation of rationals. For integer exponents, see qexpclz 14052. (Contributed by NM, 16-Dec-2005.)
((𝐴 ∈ β„š ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ β„š)
 
Theoremreexpcl 14049 Closure of exponentiation of reals. For integer exponents, see reexpclz 14053. (Contributed by NM, 14-Dec-2005.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ ℝ)
 
Theoremexpcl 14050 Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 14055. (Contributed by NM, 26-May-2005.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑𝑁) ∈ β„‚)
 
Theoremrpexpcl 14051 Closure law for integer exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) ∈ ℝ+)
 
Theoremqexpclz 14052 Closure of integer exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ β„š ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) ∈ β„š)
 
Theoremreexpclz 14053 Closure of integer exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) ∈ ℝ)
 
Theoremexpclzlem 14054 Lemma for expclz 14055. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) ∈ (β„‚ βˆ– {0}))
 
Theoremexpclz 14055 Closure law for integer exponentiation of complex numnbers. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) ∈ β„‚)
 
Theoremm1expcl2 14056 Closure of integer exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ β„€ β†’ (-1↑𝑁) ∈ {-1, 1})
 
Theoremm1expcl 14057 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ∈ β„€ β†’ (-1↑𝑁) ∈ β„€)
 
Theoremzexpcld 14058 Closure of exponentiation of integers, deduction form. (Contributed by SN, 15-Sep-2024.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (𝐴↑𝑁) ∈ β„€)
 
Theoremnn0expcli 14059 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ β„•0    &   π‘ ∈ β„•0    β‡’   (𝐴↑𝑁) ∈ β„•0
 
Theoremnn0sqcl 14060 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(𝐴 ∈ β„•0 β†’ (𝐴↑2) ∈ β„•0)
 
Theoremexpm1t 14061 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ (𝐴↑𝑁) = ((𝐴↑(𝑁 βˆ’ 1)) Β· 𝐴))
 
Theorem1exp 14062 Value of 1 raised to an integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝑁 ∈ β„€ β†’ (1↑𝑁) = 1)
 
Theoremexpeq0 14063 A positive integer power is zero if and only if its base is zero. (Contributed by NM, 23-Feb-2005.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0))
 
Theoremexpne0 14064 A positive integer power is nonzero if and only if its base is nonzero. (Contributed by NM, 6-May-2005.)
((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•) β†’ ((𝐴↑𝑁) β‰  0 ↔ 𝐴 β‰  0))
 
Theoremexpne0i 14065 An integer power is nonzero if its base is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑𝑁) β‰  0)
 
Theoremexpgt0 14066 A positive real raised to an integer power is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ β„€ ∧ 0 < 𝐴) β†’ 0 < (𝐴↑𝑁))
 
Theoremexpnegz 14067 Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑-𝑁) = (1 / (𝐴↑𝑁)))
 
Theorem0exp 14068 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ∈ β„• β†’ (0↑𝑁) = 0)
 
Theoremexpge0 14069 A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ β„•0 ∧ 0 ≀ 𝐴) β†’ 0 ≀ (𝐴↑𝑁))
 
Theoremexpge1 14070 A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ β„•0 ∧ 1 ≀ 𝐴) β†’ 1 ≀ (𝐴↑𝑁))
 
Theoremexpgt1 14071 A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ β„• ∧ 1 < 𝐴) β†’ 1 < (𝐴↑𝑁))
 
Theoremmulexp 14072 Nonnegative integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 Β· 𝐡)↑𝑁) = ((𝐴↑𝑁) Β· (𝐡↑𝑁)))
 
Theoremmulexpz 14073 Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ (𝐡 ∈ β„‚ ∧ 𝐡 β‰  0) ∧ 𝑁 ∈ β„€) β†’ ((𝐴 Β· 𝐡)↑𝑁) = ((𝐴↑𝑁) Β· (𝐡↑𝑁)))
 
Theoremexprec 14074 Integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
 
Theoremexpadd 14075 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
((𝐴 ∈ β„‚ ∧ 𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) Β· (𝐴↑𝑁)))
 
Theoremexpaddzlem 14076 Lemma for expaddz 14077. (Contributed by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ β„•) ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) Β· (𝐴↑𝑁)))
 
Theoremexpaddz 14077 Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ (𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€)) β†’ (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) Β· (𝐴↑𝑁)))
 
Theoremexpmul 14078 Product of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
((𝐴 ∈ β„‚ ∧ 𝑀 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ (𝐴↑(𝑀 Β· 𝑁)) = ((𝐴↑𝑀)↑𝑁))
 
Theoremexpmulz 14079 Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ (𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€)) β†’ (𝐴↑(𝑀 Β· 𝑁)) = ((𝐴↑𝑀)↑𝑁))
 
Theoremm1expeven 14080 Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.)
(𝑁 ∈ β„€ β†’ (-1↑(2 Β· 𝑁)) = 1)
 
Theoremexpsub 14081 Exponent subtraction law for integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) ∧ (𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€)) β†’ (𝐴↑(𝑀 βˆ’ 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁)))
 
Theoremexpp1z 14082 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) Β· 𝐴))
 
Theoremexpm1 14083 Value of a nonzero complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0 ∧ 𝑁 ∈ β„€) β†’ (𝐴↑(𝑁 βˆ’ 1)) = ((𝐴↑𝑁) / 𝐴))
 
Theoremexpdiv 14084 Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
((𝐴 ∈ β„‚ ∧ (𝐡 ∈ β„‚ ∧ 𝐡 β‰  0) ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 / 𝐡)↑𝑁) = ((𝐴↑𝑁) / (𝐡↑𝑁)))
 
Theoremsqval 14085 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴 ∈ β„‚ β†’ (𝐴↑2) = (𝐴 Β· 𝐴))
 
Theoremsqneg 14086 The square of the negative of a number. (Contributed by NM, 15-Jan-2006.)
(𝐴 ∈ β„‚ β†’ (-𝐴↑2) = (𝐴↑2))
 
Theoremsqsubswap 14087 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 βˆ’ 𝐡)↑2) = ((𝐡 βˆ’ 𝐴)↑2))
 
Theoremsqcl 14088 Closure of square. (Contributed by NM, 10-Aug-1999.)
(𝐴 ∈ β„‚ β†’ (𝐴↑2) ∈ β„‚)
 
Theoremsqmul 14089 Distribution of squaring over multiplication. (Contributed by NM, 21-Mar-2008.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 Β· 𝐡)↑2) = ((𝐴↑2) Β· (𝐡↑2)))
 
Theoremsqeq0 14090 A complex number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ β„‚ β†’ ((𝐴↑2) = 0 ↔ 𝐴 = 0))
 
Theoremsqdiv 14091 Distribution of squaring over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐡 β‰  0) β†’ ((𝐴 / 𝐡)↑2) = ((𝐴↑2) / (𝐡↑2)))
 
Theoremsqdivid 14092 The square of a nonzero complex number divided by itself equals that number. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ ((𝐴↑2) / 𝐴) = 𝐴)
 
Theoremsqne0 14093 A complex number is nonzero if and only if its square is nonzero. (Contributed by NM, 11-Mar-2006.)
(𝐴 ∈ β„‚ β†’ ((𝐴↑2) β‰  0 ↔ 𝐴 β‰  0))
 
Theoremresqcl 14094 Closure of squaring in reals. (Contributed by NM, 18-Oct-1999.)
(𝐴 ∈ ℝ β†’ (𝐴↑2) ∈ ℝ)
 
Theoremresqcld 14095 Closure of squaring in reals, deduction form. (Contributed by Mario Carneiro, 28-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐴↑2) ∈ ℝ)
 
Theoremsqgt0 14096 The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ 0 < (𝐴↑2))
 
Theoremsqn0rp 14097 The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023.)
((𝐴 ∈ ℝ ∧ 𝐴 β‰  0) β†’ (𝐴↑2) ∈ ℝ+)
 
Theoremnnsqcl 14098 The positive naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴 ∈ β„• β†’ (𝐴↑2) ∈ β„•)
 
Theoremzsqcl 14099 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴 ∈ β„€ β†’ (𝐴↑2) ∈ β„€)
 
Theoremqsqcl 14100 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(𝐴 ∈ β„š β†’ (𝐴↑2) ∈ β„š)
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