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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodrbdclem2 11601* Lemma for prodrbdc 11602. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ DECID π‘˜ ∈ 𝐴)    β‡’   ((πœ‘ ∧ 𝑁 ∈ (β„€β‰₯β€˜π‘€)) β†’ (seq𝑀( Β· , 𝐹) ⇝ 𝐢 ↔ seq𝑁( Β· , 𝐹) ⇝ 𝐢))
 
Theoremprodrbdc 11602* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘)) β†’ DECID π‘˜ ∈ 𝐴)    β‡’   (πœ‘ β†’ (seq𝑀( Β· , 𝐹) ⇝ 𝐢 ↔ seq𝑁( Β· , 𝐹) ⇝ 𝐢))
 
Theoremprodmodclem3 11603* Lemma for prodmodc 11606. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   π» = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(πΎβ€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   (πœ‘ β†’ (𝑀 ∈ β„• ∧ 𝑁 ∈ β„•))    &   (πœ‘ β†’ 𝑓:(1...𝑀)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾:(1...𝑁)–1-1-onto→𝐴)    β‡’   (πœ‘ β†’ (seq1( Β· , 𝐺)β€˜π‘€) = (seq1( Β· , 𝐻)β€˜π‘))
 
Theoremprodmodclem2a 11604* Lemma for prodmodc 11606. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   π» = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(πΎβ€˜π‘—) / π‘˜β¦Œπ΅, 1))    &   ((πœ‘ ∧ π‘˜ ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID π‘˜ ∈ 𝐴)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝐴 βŠ† (β„€β‰₯β€˜π‘€))    &   (πœ‘ β†’ 𝑓:(1...𝑁)–1-1-onto→𝐴)    &   (πœ‘ β†’ 𝐾 Isom < , < ((1...(β™―β€˜π΄)), 𝐴))    β‡’   (πœ‘ β†’ seq𝑀( Β· , 𝐹) ⇝ (seq1( Β· , 𝐺)β€˜π‘))
 
Theoremprodmodclem2 11605* Lemma for prodmodc 11606. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    β‡’   ((πœ‘ ∧ βˆƒπ‘š ∈ β„€ ((𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴) ∧ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , 𝐹) ⇝ 𝑦) ∧ seqπ‘š( Β· , 𝐹) ⇝ π‘₯))) β†’ (βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( Β· , 𝐺)β€˜π‘š)) β†’ π‘₯ = 𝑧))
 
Theoremprodmodc 11606* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.) (Modified by Jim Kingdon, 14-Apr-2024.)
𝐹 = (π‘˜ ∈ β„€ ↦ if(π‘˜ ∈ 𝐴, 𝐡, 1))    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    &   πΊ = (𝑗 ∈ β„• ↦ if(𝑗 ≀ (β™―β€˜π΄), ⦋(π‘“β€˜π‘—) / π‘˜β¦Œπ΅, 1))    β‡’   (πœ‘ β†’ βˆƒ*π‘₯(βˆƒπ‘š ∈ β„€ ((𝐴 βŠ† (β„€β‰₯β€˜π‘š) ∧ βˆ€π‘— ∈ (β„€β‰₯β€˜π‘š)DECID 𝑗 ∈ 𝐴) ∧ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)βˆƒπ‘¦(𝑦 # 0 ∧ seq𝑛( Β· , 𝐹) ⇝ 𝑦) ∧ seqπ‘š( Β· , 𝐹) ⇝ π‘₯)) ∨ βˆƒπ‘š ∈ β„• βˆƒπ‘“(𝑓:(1...π‘š)–1-1-onto→𝐴 ∧ π‘₯ = (seq1( Β· , 𝐺)β€˜π‘š))))
 
Theoremzproddc 11607* Series product with index set a subset of the upper integers. (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‘ β†’ 𝐴 βŠ† 𝑍)    &   (πœ‘ β†’ βˆ€π‘— ∈ 𝑍 DECID 𝑗 ∈ 𝐴)    &   ((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 = ( ⇝ β€˜seq𝑀( Β· , 𝐹)))
 
Theoremiprodap 11608* 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 11609* 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 11610* 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 11611* 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 11612* 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 11613 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
βˆπ‘˜ ∈ βˆ… 𝐴 = 1
 
Theoremprod1dc 11614* 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 11615* 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 11616* 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 11617* 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 11618* 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 11619* 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 11620* A product of a singleton is the term. A version of prodsn 11621 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 11621* 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 11622* 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 11623 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 11624* Split a finite product into two parts. New proofs should use fprodsplit 11625 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 11625* 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 11626* 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 11627* 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 11628* 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 11629* 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 11630* 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 11631* 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 11632* Multiply in an additional term in a finite product. See also fprodsplitsn 11661 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 11633* 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 11634* 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 11635* 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 11636* 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 11637* 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 11638* 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 11639* 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 11640* 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 11641* Finite product closure lemma. A version of fprodcllem 11634 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 11642* Closure of a finite product of real numbers. A version of fprodrecl 11636 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 11643* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝐴 ∈ β„•0 β†’ (!β€˜π΄) = βˆπ‘˜ ∈ (1...𝐴)π‘˜)
 
Theoremfprodabs 11644* 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 11645* 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 11646* 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 11647* 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 11648* 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 11649* 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 11650* Lemma for fprod2d 11651- 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 11651* Write a double product as a product over a two-dimensional region. Compare fsum2d 11463. (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 11652* 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 11653* 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 11654* 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 11655* 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 11656* Two ways to express "the product of 𝐴(𝑗, π‘˜) over the triangular region 𝑀 ≀ 𝑗, 𝑀 ≀ π‘˜, 𝑗 + π‘˜ ≀ 𝑁. Compare fisum0diag 11469. (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 11657* 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 11658* 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 11659* The quotient of two finite products. A version of fproddivap 11658 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 11660* Split a finite product into two parts. A version of fprodsplit 11625 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 11661* Separate out a term in a finite product. See also fprodunsn 11632 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 11662* 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 11663* Closure of a finite product of complex numbers. A version of fprodcl 11635 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 11664* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11649 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 11665* 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 11666* 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 11667* 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 11668* 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 11669* 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 11670 Extend class notation to include the exponential function.
class exp
 
Syntaxceu 11671 Extend class notation to include Euler's constant e = 2.71828....
class e
 
Syntaxcsin 11672 Extend class notation to include the sine function.
class sin
 
Syntaxccos 11673 Extend class notation to include the cosine function.
class cos
 
Syntaxctan 11674 Extend class notation to include the tangent function.
class tan
 
Syntaxcpi 11675 Extend class notation to include the constant pi, Ο€ = 3.14159....
class Ο€
 
Definitiondf-ef 11676* Define the exponential function. Its value at the complex number 𝐴 is (expβ€˜π΄) and is called the "exponential of 𝐴"; see efval 11689. (Contributed by NM, 14-Mar-2005.)
exp = (π‘₯ ∈ β„‚ ↦ Ξ£π‘˜ ∈ β„•0 ((π‘₯β†‘π‘˜) / (!β€˜π‘˜)))
 
Definitiondf-e 11677 Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (expβ€˜1)
 
Definitiondf-sin 11678 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 11679 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (π‘₯ ∈ β„‚ ↦ (((expβ€˜(i Β· π‘₯)) + (expβ€˜(-i Β· π‘₯))) / 2))
 
Definitiondf-tan 11680 Define the tangent function. We define it this way for cmpt 4079, 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 11681 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 11682 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 11683 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 11684 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 11685* 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 11686* Lemma for efcl 11692. The series that defines the exponential function converges. The ratio test cvgratgt0 11561 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 11687* Lemma for efcl 11692. 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 11688* 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 11689* 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 11690 Value of Euler's constant e = 2.71828.... (Contributed by Steve Rodriguez, 5-Mar-2006.)
e = Ξ£π‘˜ ∈ β„•0 (1 / (!β€˜π‘˜))
 
Theoremeff 11691 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 11692 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 11693* 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 11694* 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 11695* 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 11696 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 11697 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 11698 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 11699 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 11700 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
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