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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiprodap 11601* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ βˆƒπ‘› ∈ 𝑍 βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , 𝐹) ⇝ 𝑦))    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝑍 𝐡 = ( ⇝ β€˜seq𝑀( Β· , 𝐹)))
 
Theoremzprodap0 11602* Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑋 # 0)    &   (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)    &   (πœ‘ β†’ βˆ€π‘— ∈ 𝑍 DECID 𝑗 ∈ 𝐴)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑍)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = 𝑋)
 
Theoremiprodap0 11603* Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑋 # 0)    &   (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ 𝑋)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ (πΉβ€˜π‘˜) = 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝑍 𝐡 = 𝑋)
 
4.8.10.4  Finite products
 
Theoremfprodseq 11604* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
(π‘˜ = (πΉβ€˜π‘›) β†’ 𝐡 = 𝐢)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑀)–1-1-onto→𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ 𝑛 ∈ (1...𝑀)) β†’ (πΊβ€˜π‘›) = 𝐢)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = (seq1( Β· , (𝑛 ∈ β„• ↦ if(𝑛 ≀ 𝑀, (πΊβ€˜π‘›), 1)))β€˜π‘€))
 
Theoremfprodntrivap 11605* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   (πœ‘ β†’ 𝐴 βŠ† (𝑀...𝑁))    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ 𝑍 βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , (π‘˜ ∈ 𝑍 ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))) ⇝ 𝑦))
 
Theoremprod0 11606 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
βˆπ‘˜ ∈ βˆ… 𝐴 = 1
 
Theoremprod1dc 11607* Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)
(((𝑀 ∈ β„€ ∧ 𝐴 βŠ† (β„€β‰₯β€˜π‘€) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) β†’ βˆπ‘˜ ∈ 𝐴 1 = 1)
 
Theoremprodfct 11608* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
(βˆ€π‘˜ ∈ 𝐴 𝐡 ∈ β„‚ β†’ βˆπ‘— ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘—) = βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodf1o 11609* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
(π‘˜ = 𝐺 β†’ 𝐡 = 𝐷)    &   (πœ‘ β†’ 𝐢 ∈ Fin)    &   (πœ‘ β†’ 𝐹:𝐢–1-1-onto→𝐴)    &   ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ (πΉβ€˜π‘›) = 𝐺)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷)
 
Theoremprodssdc 11610* Change the index set to a subset in an upper integer product. (Contributed by Scott Fenton, 11-Dec-2017.) (Revised by Jim Kingdon, 6-Aug-2024.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , (π‘˜ ∈ (β„€β‰₯β€˜π‘€) ↦ if(π‘˜ ∈ 𝐡, 𝐢, 1))) ⇝ 𝑦))    &   (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐴)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)    &   (πœ‘ β†’ 𝐡 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘€)DECID 𝑗 ∈ 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
 
Theoremfprodssdc 11611* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 1)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐢 = βˆπ‘˜ ∈ 𝐡 𝐢)
 
Theoremfprodmul 11612* The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (𝐡 Β· 𝐢) = (βˆπ‘˜ ∈ 𝐴 𝐡 Β· βˆπ‘˜ ∈ 𝐴 𝐢))
 
Theoremprodsnf 11613* A product of a singleton is the term. A version of prodsn 11614 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜π΅    &   (π‘˜ = 𝑀 β†’ 𝐴 = 𝐡)    β‡’   ((𝑀 ∈ 𝑉 ∧ 𝐡 ∈ β„‚) β†’ βˆπ‘˜ ∈ {𝑀}𝐴 = 𝐡)
 
Theoremprodsn 11614* A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)
(π‘˜ = 𝑀 β†’ 𝐴 = 𝐡)    β‡’   ((𝑀 ∈ 𝑉 ∧ 𝐡 ∈ β„‚) β†’ βˆπ‘˜ ∈ {𝑀}𝐴 = 𝐡)
 
Theoremfprod1 11615* A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)
(π‘˜ = 𝑀 β†’ 𝐴 = 𝐡)    β‡’   ((𝑀 ∈ β„€ ∧ 𝐡 ∈ β„‚) β†’ βˆπ‘˜ ∈ (𝑀...𝑀)𝐴 = 𝐡)
 
Theoremclimprod1 11616 The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    β‡’   (πœ‘ β†’ seq𝑀( Β· , (𝑍 Γ— {1})) ⇝ 1)
 
Theoremfprodsplitdc 11617* Split a finite product into two parts. New proofs should use fprodsplit 11618 which is the same but with one fewer hypothesis. (Contributed by Scott Fenton, 16-Dec-2017.) (New usage is discouraged.)
(πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   (πœ‘ β†’ π‘ˆ ∈ Fin)    &   (πœ‘ β†’ βˆ€π‘— ∈ π‘ˆ DECID 𝑗 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ π‘ˆ 𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· βˆπ‘˜ ∈ 𝐡 𝐢))
 
Theoremfprodsplit 11618* Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)
(πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   (πœ‘ β†’ π‘ˆ ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ π‘ˆ 𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· βˆπ‘˜ ∈ 𝐡 𝐢))
 
Theoremfprodm1 11619* Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ 𝐴 ∈ β„‚)    &   (π‘˜ = 𝑁 β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝑀...𝑁)𝐴 = (βˆπ‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))𝐴 Β· 𝐡))
 
Theoremfprod1p 11620* Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ 𝐴 ∈ β„‚)    &   (π‘˜ = 𝑀 β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝑀...𝑁)𝐴 = (𝐡 Β· βˆπ‘˜ ∈ ((𝑀 + 1)...𝑁)𝐴))
 
Theoremfprodp1 11621* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 + 1))) β†’ 𝐴 ∈ β„‚)    &   (π‘˜ = (𝑁 + 1) β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝑀...(𝑁 + 1))𝐴 = (βˆπ‘˜ ∈ (𝑀...𝑁)𝐴 Β· 𝐡))
 
Theoremfprodm1s 11622* Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...𝑁)) β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝑀...𝑁)𝐴 = (βˆπ‘˜ ∈ (𝑀...(𝑁 βˆ’ 1))𝐴 Β· ⦋𝑁 / π‘˜β¦Œπ΄))
 
Theoremfprodp1s 11623* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
(πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))    &   ((πœ‘ ∧ π‘˜ ∈ (𝑀...(𝑁 + 1))) β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝑀...(𝑁 + 1))𝐴 = (βˆπ‘˜ ∈ (𝑀...𝑁)𝐴 Β· ⦋(𝑁 + 1) / π‘˜β¦Œπ΄))
 
Theoremprodsns 11624* A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)
((𝑀 ∈ 𝑉 ∧ ⦋𝑀 / π‘˜β¦Œπ΄ ∈ β„‚) β†’ βˆπ‘˜ ∈ {𝑀}𝐴 = ⦋𝑀 / π‘˜β¦Œπ΄)
 
Theoremfprodunsn 11625* Multiply in an additional term in a finite product. See also fprodsplitsn 11654 which is the same but with a β„²π‘˜πœ‘ hypothesis in place of the distinct variable condition between πœ‘ and π‘˜. (Contributed by Jim Kingdon, 16-Aug-2024.)
β„²π‘˜π·    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (π‘˜ = 𝐡 β†’ 𝐢 = 𝐷)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝐴 βˆͺ {𝐡})𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· 𝐷))
 
Theoremfprodcl2lem 11626* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 β‰  βˆ…)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ 𝑆)
 
Theoremfprodcllem 11627* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝑆 βŠ† β„‚)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 1 ∈ 𝑆)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ 𝑆)
 
Theoremfprodcl 11628* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ β„‚)
 
Theoremfprodrecl 11629* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ ℝ)
 
Theoremfprodzcl 11630* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„€)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ β„€)
 
Theoremfprodnncl 11631* Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„•)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ β„•)
 
Theoremfprodrpcl 11632* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ ℝ+)
 
Theoremfprodnn0cl 11633* Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„•0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ β„•0)
 
Theoremfprodcllemf 11634* Finite product closure lemma. A version of fprodcllem 11627 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝑆 βŠ† β„‚)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯ Β· 𝑦) ∈ 𝑆)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ 𝑆)    &   (πœ‘ β†’ 1 ∈ 𝑆)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ 𝑆)
 
Theoremfprodreclf 11635* Closure of a finite product of real numbers. A version of fprodrecl 11629 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ ℝ)
 
Theoremfprodfac 11636* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝐴 ∈ β„•0 β†’ (!β€˜π΄) = βˆπ‘˜ ∈ (1...𝐴)π‘˜)
 
Theoremfprodabs 11637* The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐴 ∈ β„‚)    β‡’   (πœ‘ β†’ (absβ€˜βˆπ‘˜ ∈ (𝑀...𝑁)𝐴) = βˆπ‘˜ ∈ (𝑀...𝑁)(absβ€˜π΄))
 
Theoremfprodeq0 11638* Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
𝑍 = (β„€β‰₯β€˜π‘€)    &   (πœ‘ β†’ 𝑁 ∈ 𝑍)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑍) β†’ 𝐴 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ = 𝑁) β†’ 𝐴 = 0)    β‡’   ((πœ‘ ∧ 𝐾 ∈ (β„€β‰₯β€˜π‘)) β†’ βˆπ‘˜ ∈ (𝑀...𝐾)𝐴 = 0)
 
Theoremfprodshft 11639* Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   ((πœ‘ ∧ 𝑗 ∈ (𝑀...𝑁)) β†’ 𝐴 ∈ β„‚)    &   (𝑗 = (π‘˜ βˆ’ 𝐾) β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ (𝑀...𝑁)𝐴 = βˆπ‘˜ ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐡)
 
Theoremfprodrev 11640* Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   ((πœ‘ ∧ 𝑗 ∈ (𝑀...𝑁)) β†’ 𝐴 ∈ β„‚)    &   (𝑗 = (𝐾 βˆ’ π‘˜) β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ (𝑀...𝑁)𝐴 = βˆπ‘˜ ∈ ((𝐾 βˆ’ 𝑁)...(𝐾 βˆ’ 𝑀))𝐡)
 
Theoremfprodconst 11641* The product of constant terms (π‘˜ is not free in 𝐡). (Contributed by Scott Fenton, 12-Jan-2018.)
((𝐴 ∈ Fin ∧ 𝐡 ∈ β„‚) β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = (𝐡↑(β™―β€˜π΄)))
 
Theoremfprodap0 11642* A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 # 0)
 
Theoremfprod2dlemstep 11643* Lemma for fprod2d 11644- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
(𝑧 = βŸ¨π‘—, π‘˜βŸ© β†’ 𝐷 = 𝐢)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ 𝑗 ∈ 𝐴) β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ (𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡)) β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ Β¬ 𝑦 ∈ π‘₯)    &   (πœ‘ β†’ (π‘₯ βˆͺ {𝑦}) βŠ† 𝐴)    &   (πœ‘ β†’ π‘₯ ∈ Fin)    &   (πœ“ ↔ βˆπ‘— ∈ π‘₯ βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘§ ∈ βˆͺ 𝑗 ∈ π‘₯ ({𝑗} Γ— 𝐡)𝐷)    β‡’   ((πœ‘ ∧ πœ“) β†’ βˆπ‘— ∈ (π‘₯ βˆͺ {𝑦})βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘§ ∈ βˆͺ 𝑗 ∈ (π‘₯ βˆͺ {𝑦})({𝑗} Γ— 𝐡)𝐷)
 
Theoremfprod2d 11644* Write a double product as a product over a two-dimensional region. Compare fsum2d 11456. (Contributed by Scott Fenton, 30-Jan-2018.)
(𝑧 = βŸ¨π‘—, π‘˜βŸ© β†’ 𝐷 = 𝐢)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ 𝑗 ∈ 𝐴) β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ (𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡)) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ 𝐴 βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘§ ∈ βˆͺ 𝑗 ∈ 𝐴 ({𝑗} Γ— 𝐡)𝐷)
 
Theoremfprodxp 11645* Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
(𝑧 = βŸ¨π‘—, π‘˜βŸ© β†’ 𝐷 = 𝐢)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ (𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡)) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ 𝐴 βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘§ ∈ (𝐴 Γ— 𝐡)𝐷)
 
Theoremfprodcnv 11646* Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
(π‘₯ = βŸ¨π‘—, π‘˜βŸ© β†’ 𝐡 = 𝐷)    &   (𝑦 = βŸ¨π‘˜, π‘—βŸ© β†’ 𝐢 = 𝐷)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ Rel 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ ∏π‘₯ ∈ 𝐴 𝐡 = βˆπ‘¦ ∈ β—‘ 𝐴𝐢)
 
Theoremfprodcom2fi 11647* Interchange order of multiplication. Note that 𝐡(𝑗) and 𝐷(π‘˜) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐢 ∈ Fin)    &   ((πœ‘ ∧ 𝑗 ∈ 𝐴) β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐢) β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ ((𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡) ↔ (π‘˜ ∈ 𝐢 ∧ 𝑗 ∈ 𝐷)))    &   ((πœ‘ ∧ (𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡)) β†’ 𝐸 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ 𝐴 βˆπ‘˜ ∈ 𝐡 𝐸 = βˆπ‘˜ ∈ 𝐢 βˆπ‘— ∈ 𝐷 𝐸)
 
Theoremfprodcom 11648* Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   ((πœ‘ ∧ (𝑗 ∈ 𝐴 ∧ π‘˜ ∈ 𝐡)) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ 𝐴 βˆπ‘˜ ∈ 𝐡 𝐢 = βˆπ‘˜ ∈ 𝐡 βˆπ‘— ∈ 𝐴 𝐢)
 
Theoremfprod0diagfz 11649* Two ways to express "the product of 𝐴(𝑗, π‘˜) over the triangular region 𝑀 ≀ 𝑗, 𝑀 ≀ π‘˜, 𝑗 + π‘˜ ≀ 𝑁. Compare fisum0diag 11462. (Contributed by Scott Fenton, 2-Feb-2018.)
((πœ‘ ∧ (𝑗 ∈ (0...𝑁) ∧ π‘˜ ∈ (0...(𝑁 βˆ’ 𝑗)))) β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ βˆπ‘— ∈ (0...𝑁)βˆπ‘˜ ∈ (0...(𝑁 βˆ’ 𝑗))𝐴 = βˆπ‘˜ ∈ (0...𝑁)βˆπ‘— ∈ (0...(𝑁 βˆ’ π‘˜))𝐴)
 
Theoremfprodrec 11650* The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (1 / 𝐡) = (1 / βˆπ‘˜ ∈ 𝐴 𝐡))
 
Theoremfproddivap 11651* The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (𝐡 / 𝐢) = (βˆπ‘˜ ∈ 𝐴 𝐡 / βˆπ‘˜ ∈ 𝐴 𝐢))
 
Theoremfproddivapf 11652* The quotient of two finite products. A version of fproddivap 11651 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 (𝐡 / 𝐢) = (βˆπ‘˜ ∈ 𝐴 𝐡 / βˆπ‘˜ ∈ 𝐴 𝐢))
 
Theoremfprodsplitf 11653* Split a finite product into two parts. A version of fprodsplit 11618 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ (𝐴 ∩ 𝐡) = βˆ…)    &   (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))    &   (πœ‘ β†’ π‘ˆ ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ π‘ˆ 𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· βˆπ‘˜ ∈ 𝐡 𝐢))
 
Theoremfprodsplitsn 11654* Separate out a term in a finite product. See also fprodunsn 11625 which is the same but with a distinct variable condition in place of β„²π‘˜πœ‘. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   β„²π‘˜π·    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)    &   (π‘˜ = 𝐡 β†’ 𝐢 = 𝐷)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ (𝐴 βˆͺ {𝐡})𝐢 = (βˆπ‘˜ ∈ 𝐴 𝐢 Β· 𝐷))
 
Theoremfprodsplit1f 11655* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ β„²π‘˜π·)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ = 𝐢) β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = (𝐷 Β· βˆπ‘˜ ∈ (𝐴 βˆ– {𝐢})𝐡))
 
Theoremfprodclf 11656* Closure of a finite product of complex numbers. A version of fprodcl 11628 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ∈ β„‚)
 
Theoremfprodap0f 11657* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11642 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 # 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 # 0)
 
Theoremfprodge0 11658* If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 0 ≀ 𝐡)    β‡’   (πœ‘ β†’ 0 ≀ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodeq0g 11659* Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ = 𝐢) β†’ 𝐡 = 0)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = 0)
 
Theoremfprodge1 11660* If all of the terms of a finite product are greater than or equal to 1, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 1 ≀ 𝐡)    β‡’   (πœ‘ β†’ 1 ≀ βˆπ‘˜ ∈ 𝐴 𝐡)
 
Theoremfprodle 11661* If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
β„²π‘˜πœ‘    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 0 ≀ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ ℝ)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ≀ 𝐢)    β‡’   (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 ≀ βˆπ‘˜ ∈ 𝐴 𝐢)
 
Theoremfprodmodd 11662* If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„€)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (𝐡 mod 𝑀) = (𝐢 mod 𝑀))    β‡’   (πœ‘ β†’ (βˆπ‘˜ ∈ 𝐴 𝐡 mod 𝑀) = (βˆπ‘˜ ∈ 𝐴 𝐢 mod 𝑀))
 
4.9  Elementary trigonometry
 
4.9.1  The exponential, sine, and cosine functions
 
Syntaxce 11663 Extend class notation to include the exponential function.
class exp
 
Syntaxceu 11664 Extend class notation to include Euler's constant e = 2.71828....
class e
 
Syntaxcsin 11665 Extend class notation to include the sine function.
class sin
 
Syntaxccos 11666 Extend class notation to include the cosine function.
class cos
 
Syntaxctan 11667 Extend class notation to include the tangent function.
class tan
 
Syntaxcpi 11668 Extend class notation to include the constant pi, Ο€ = 3.14159....
class Ο€
 
Definitiondf-ef 11669* Define the exponential function. Its value at the complex number 𝐴 is (expβ€˜π΄) and is called the "exponential of 𝐴"; see efval 11682. (Contributed by NM, 14-Mar-2005.)
exp = (π‘₯ ∈ β„‚ ↦ Ξ£π‘˜ ∈ β„•0 ((π‘₯β†‘π‘˜) / (!β€˜π‘˜)))
 
Definitiondf-e 11670 Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (expβ€˜1)
 
Definitiondf-sin 11671 Define the sine function. (Contributed by NM, 14-Mar-2005.)
sin = (π‘₯ ∈ β„‚ ↦ (((expβ€˜(i Β· π‘₯)) βˆ’ (expβ€˜(-i Β· π‘₯))) / (2 Β· i)))
 
Definitiondf-cos 11672 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (π‘₯ ∈ β„‚ ↦ (((expβ€˜(i Β· π‘₯)) + (expβ€˜(-i Β· π‘₯))) / 2))
 
Definitiondf-tan 11673 Define the tangent function. We define it this way for cmpt 4076, which requires the form (π‘₯ ∈ 𝐴 ↦ 𝐡). (Contributed by Mario Carneiro, 14-Mar-2014.)
tan = (π‘₯ ∈ (β—‘cos β€œ (β„‚ βˆ– {0})) ↦ ((sinβ€˜π‘₯) / (cosβ€˜π‘₯)))
 
Definitiondf-pi 11674 Define the constant pi, Ο€ = 3.14159..., which is the smallest positive number whose sine is zero. Definition of Ο€ in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
Ο€ = inf((ℝ+ ∩ (β—‘sin β€œ {0})), ℝ, < )
 
Theoremeftcl 11675 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
((𝐴 ∈ β„‚ ∧ 𝐾 ∈ β„•0) β†’ ((𝐴↑𝐾) / (!β€˜πΎ)) ∈ β„‚)
 
Theoremreeftcl 11676 The terms of the series expansion of the exponential function at a real number are real. (Contributed by Paul Chapman, 15-Jan-2008.)
((𝐴 ∈ ℝ ∧ 𝐾 ∈ β„•0) β†’ ((𝐴↑𝐾) / (!β€˜πΎ)) ∈ ℝ)
 
Theoremeftabs 11677 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
((𝐴 ∈ β„‚ ∧ 𝐾 ∈ β„•0) β†’ (absβ€˜((𝐴↑𝐾) / (!β€˜πΎ))) = (((absβ€˜π΄)↑𝐾) / (!β€˜πΎ)))
 
Theoremeftvalcn 11678* The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝑁 ∈ β„•0) β†’ (πΉβ€˜π‘) = ((𝐴↑𝑁) / (!β€˜π‘)))
 
Theoremefcllemp 11679* Lemma for efcl 11685. The series that defines the exponential function converges. The ratio test cvgratgt0 11554 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐾 ∈ β„•)    &   (πœ‘ β†’ (2 Β· (absβ€˜π΄)) < 𝐾)    β‡’   (πœ‘ β†’ seq0( + , 𝐹) ∈ dom ⇝ )
 
Theoremefcllem 11680* Lemma for efcl 11685. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ seq0( + , 𝐹) ∈ dom ⇝ )
 
Theoremef0lem 11681* The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 = 0 β†’ seq0( + , 𝐹) ⇝ 1)
 
Theoremefval 11682* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) = Ξ£π‘˜ ∈ β„•0 ((π΄β†‘π‘˜) / (!β€˜π‘˜)))
 
Theoremesum 11683 Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
e = Ξ£π‘˜ ∈ β„•0 (1 / (!β€˜π‘˜))
 
Theoremeff 11684 Domain and codomain of the exponential function. (Contributed by Paul Chapman, 22-Oct-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
exp:β„‚βŸΆβ„‚
 
Theoremefcl 11685 Closure law for the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) ∈ β„‚)
 
Theoremefval2 11686* Value of the exponential function. (Contributed by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ (expβ€˜π΄) = Ξ£π‘˜ ∈ β„•0 (πΉβ€˜π‘˜))
 
Theoremefcvg 11687* The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    β‡’   (𝐴 ∈ β„‚ β†’ seq0( + , 𝐹) ⇝ (expβ€˜π΄))
 
Theoremefcvgfsum 11688* Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ Ξ£π‘˜ ∈ (0...𝑛)((π΄β†‘π‘˜) / (!β€˜π‘˜)))    β‡’   (𝐴 ∈ β„‚ β†’ 𝐹 ⇝ (expβ€˜π΄))
 
Theoremreefcl 11689 The exponential function is real if its argument is real. (Contributed by NM, 27-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
(𝐴 ∈ ℝ β†’ (expβ€˜π΄) ∈ ℝ)
 
Theoremreefcld 11690 The exponential function is real if its argument is real. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    β‡’   (πœ‘ β†’ (expβ€˜π΄) ∈ ℝ)
 
Theoremere 11691 Euler's constant e = 2.71828... is a real number. (Contributed by NM, 19-Mar-2005.) (Revised by Steve Rodriguez, 8-Mar-2006.)
e ∈ ℝ
 
Theoremege2le3 11692 Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 20-Mar-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2014.)
𝐹 = (𝑛 ∈ β„• ↦ (2 Β· ((1 / 2)↑𝑛)))    &   πΊ = (𝑛 ∈ β„•0 ↦ (1 / (!β€˜π‘›)))    β‡’   (2 ≀ e ∧ e ≀ 3)
 
Theoremef0 11693 Value of the exponential function at 0. Equation 2 of [Gleason] p. 308. (Contributed by Steve Rodriguez, 27-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)
(expβ€˜0) = 1
 
Theoremefcj 11694 The exponential of a complex conjugate. Equation 3 of [Gleason] p. 308. (Contributed by NM, 29-Apr-2005.) (Revised by Mario Carneiro, 28-Apr-2014.)
(𝐴 ∈ β„‚ β†’ (expβ€˜(βˆ—β€˜π΄)) = (βˆ—β€˜(expβ€˜π΄)))
 
Theoremefaddlem 11695* Lemma for efadd 11696 (exponential function addition law). (Contributed by Mario Carneiro, 29-Apr-2014.)
𝐹 = (𝑛 ∈ β„•0 ↦ ((𝐴↑𝑛) / (!β€˜π‘›)))    &   πΊ = (𝑛 ∈ β„•0 ↦ ((𝐡↑𝑛) / (!β€˜π‘›)))    &   π» = (𝑛 ∈ β„•0 ↦ (((𝐴 + 𝐡)↑𝑛) / (!β€˜π‘›)))    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (expβ€˜(𝐴 + 𝐡)) = ((expβ€˜π΄) Β· (expβ€˜π΅)))
 
Theoremefadd 11696 Sum of exponents law for exponential function. (Contributed by NM, 10-Jan-2006.) (Proof shortened by Mario Carneiro, 29-Apr-2014.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (expβ€˜(𝐴 + 𝐡)) = ((expβ€˜π΄) Β· (expβ€˜π΅)))
 
Theoremefcan 11697 Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.)
(𝐴 ∈ β„‚ β†’ ((expβ€˜π΄) Β· (expβ€˜-𝐴)) = 1)
 
Theoremefap0 11698 The exponential of a complex number is apart from zero. (Contributed by Jim Kingdon, 12-Dec-2022.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) # 0)
 
Theoremefne0 11699 The exponential of a complex number is nonzero. Corollary 15-4.3 of [Gleason] p. 309. The same result also holds with not equal replaced by apart, as seen at efap0 11698 (which will be more useful in most contexts). (Contributed by NM, 13-Jan-2006.) (Revised by Mario Carneiro, 29-Apr-2014.)
(𝐴 ∈ β„‚ β†’ (expβ€˜π΄) β‰  0)
 
Theoremefneg 11700 The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014.)
(𝐴 ∈ β„‚ β†’ (expβ€˜-𝐴) = (1 / (expβ€˜π΄)))
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