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Theorem List for Intuitionistic Logic Explorer - 11601-11700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprodeq2i 11601* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝑘𝐴𝐵 = 𝐶)       𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶
 
Theoremprodeq12i 11602* Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐴 = 𝐵    &   (𝑘𝐴𝐶 = 𝐷)       𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷
 
Theoremprodeq1d 11603* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 
Theoremprodeq2d 11604* Equality deduction for product. Note that unlike prodeq2dv 11605, 𝑘 may occur in 𝜑. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
 
Theoremprodeq2dv 11605* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
 
Theoremprodeq2sdv 11606* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
 
Theorem2cprodeq2dv 11607* Equality deduction for double product. (Contributed by Scott Fenton, 4-Dec-2017.)
((𝜑𝑗𝐴𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑗𝐴𝑘𝐵 𝐷)
 
Theoremprodeq12dv 11608* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)
 
Theoremprodeq12rdv 11609* Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐵) → 𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)
 
Theoremprodrbdclem 11610* Lemma for prodrbdc 11613. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 4-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))       ((𝜑𝐴 ⊆ (ℤ𝑁)) → (seq𝑀( · , 𝐹) ↾ (ℤ𝑁)) = seq𝑁( · , 𝐹))
 
Theoremfproddccvg 11611* The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘𝑁))
 
Theoremprodrbdclem2 11612* Lemma for prodrbdc 11613. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → DECID 𝑘𝐴)       ((𝜑𝑁 ∈ (ℤ𝑀)) → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶))
 
Theoremprodrbdc 11613* Rebase the starting point of a product. (Contributed by Scott Fenton, 4-Dec-2017.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝐴 ⊆ (ℤ𝑁))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   ((𝜑𝑘 ∈ (ℤ𝑁)) → DECID 𝑘𝐴)       (𝜑 → (seq𝑀( · , 𝐹) ⇝ 𝐶 ↔ seq𝑁( · , 𝐹) ⇝ 𝐶))
 
Theoremprodmodclem3 11614* Lemma for prodmodc 11617. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1))    &   (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))    &   (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)    &   (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)       (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
 
Theoremprodmodclem2a 11615* Lemma for prodmodc 11617. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))    &   𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝐾𝑗) / 𝑘𝐵, 1))    &   ((𝜑𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   (𝜑𝑓:(1...𝑁)–1-1-onto𝐴)    &   (𝜑𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))       (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))
 
Theoremprodmodclem2 11616* Lemma for prodmodc 11617. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 13-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))       ((𝜑 ∧ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( · , 𝐺)‘𝑚)) → 𝑥 = 𝑧))
 
Theoremprodmodc 11617* A product has at most one limit. (Contributed by Scott Fenton, 4-Dec-2017.) (Modified by Jim Kingdon, 14-Apr-2024.)
𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), (𝑓𝑗) / 𝑘𝐵, 1))       (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦) ∧ seq𝑚( · , 𝐹) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , 𝐺)‘𝑚))))
 
Theoremzproddc 11618* Series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   (𝜑𝐴𝑍)    &   (𝜑 → ∀𝑗𝑍 DECID 𝑗𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
 
Theoremiprodap 11619* Series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 5-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , 𝐹) ⇝ 𝑦))    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 𝐵 = ( ⇝ ‘seq𝑀( · , 𝐹)))
 
Theoremzprodap0 11620* Nonzero series product with index set a subset of the upper integers. (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑋 # 0)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   (𝜑 → ∀𝑗𝑍 DECID 𝑗𝐴)    &   (𝜑𝐴𝑍)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = if(𝑘𝐴, 𝐵, 1))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = 𝑋)
 
Theoremiprodap0 11621* Nonzero series product with an upper integer index set (i.e. an infinite product.) (Contributed by Scott Fenton, 6-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑋 # 0)    &   (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝑍 𝐵 = 𝑋)
 
4.8.10.4  Finite products
 
Theoremfprodseq 11622* The value of a product over a nonempty finite set. (Contributed by Scott Fenton, 6-Dec-2017.) (Revised by Jim Kingdon, 15-Jul-2024.)
(𝑘 = (𝐹𝑛) → 𝐵 = 𝐶)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)–1-1-onto𝐴)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑛 ∈ (1...𝑀)) → (𝐺𝑛) = 𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑀, (𝐺𝑛), 1)))‘𝑀))
 
Theoremfprodntrivap 11623* A non-triviality lemma for finite sequences. (Contributed by Scott Fenton, 16-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))       (𝜑 → ∃𝑛𝑍𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘𝑍 ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦))
 
Theoremprod0 11624 A product over the empty set is one. (Contributed by Scott Fenton, 5-Dec-2017.)
𝑘 ∈ ∅ 𝐴 = 1
 
Theoremprod1dc 11625* 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 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘𝐴 1 = 1)
 
Theoremprodfct 11626* A lemma to facilitate conversions from the function form to the class-variable form of a product. (Contributed by Scott Fenton, 7-Dec-2017.)
(∀𝑘𝐴 𝐵 ∈ ℂ → ∏𝑗𝐴 ((𝑘𝐴𝐵)‘𝑗) = ∏𝑘𝐴 𝐵)
 
Theoremfprodf1o 11627* Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
(𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 = ∏𝑛𝐶 𝐷)
 
Theoremprodssdc 11628* 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.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → ∃𝑛 ∈ (ℤ𝑀)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ (ℤ𝑀) ↦ if(𝑘𝐵, 𝐶, 1))) ⇝ 𝑦))    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)    &   (𝜑𝐵 ⊆ (ℤ𝑀))    &   (𝜑 → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐵)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 
Theoremfprodssdc 11629* Change the index set to a subset in a finite sum. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝐴𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝜑 → ∀𝑗𝐵 DECID 𝑗𝐴)    &   ((𝜑𝑘 ∈ (𝐵𝐴)) → 𝐶 = 1)    &   (𝜑𝐵 ∈ Fin)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
 
Theoremfprodmul 11630* The product of two finite products. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 (𝐵 · 𝐶) = (∏𝑘𝐴 𝐵 · ∏𝑘𝐴 𝐶))
 
Theoremprodsnf 11631* A product of a singleton is the term. A version of prodsn 11632 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝐵    &   (𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
 
Theoremprodsn 11632* A product of a singleton is the term. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀𝑉𝐵 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝐵)
 
Theoremfprod1 11633* A finite product of only one term is the term itself. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝑘 = 𝑀𝐴 = 𝐵)       ((𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ (𝑀...𝑀)𝐴 = 𝐵)
 
Theoremclimprod1 11634 The limit of a product over one. (Contributed by Scott Fenton, 15-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → seq𝑀( · , (𝑍 × {1})) ⇝ 1)
 
Theoremfprodsplitdc 11635* Split a finite product into two parts. New proofs should use fprodsplit 11636 which is the same but with one fewer hypothesis. (Contributed by Scott Fenton, 16-Dec-2017.) (New usage is discouraged.)
(𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   (𝜑 → ∀𝑗𝑈 DECID 𝑗𝐴)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑘𝑈 𝐶 = (∏𝑘𝐴 𝐶 · ∏𝑘𝐵 𝐶))
 
Theoremfprodsplit 11636* Split a finite product into two parts. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑘𝑈 𝐶 = (∏𝑘𝐴 𝐶 · ∏𝑘𝐵 𝐶))
 
Theoremfprodm1 11637* Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑁𝐴 = 𝐵)       (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝐵))
 
Theoremfprod1p 11638* Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = 𝑀𝐴 = 𝐵)       (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (𝐵 · ∏𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
 
Theoremfprodp1 11639* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · 𝐵))
 
Theoremfprodm1s 11640* Separate out the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (∏𝑘 ∈ (𝑀...(𝑁 − 1))𝐴 · 𝑁 / 𝑘𝐴))
 
Theoremfprodp1s 11641* Multiply in the last term in a finite product. (Contributed by Scott Fenton, 27-Dec-2017.)
(𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝑁 + 1))) → 𝐴 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · (𝑁 + 1) / 𝑘𝐴))
 
Theoremprodsns 11642* A product of the singleton is the term. (Contributed by Scott Fenton, 25-Dec-2017.)
((𝑀𝑉𝑀 / 𝑘𝐴 ∈ ℂ) → ∏𝑘 ∈ {𝑀}𝐴 = 𝑀 / 𝑘𝐴)
 
Theoremfprodunsn 11643* Multiply in an additional term in a finite product. See also fprodsplitsn 11672 which is the same but with a 𝑘𝜑 hypothesis in place of the distinct variable condition between 𝜑 and 𝑘. (Contributed by Jim Kingdon, 16-Aug-2024.)
𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)       (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘𝐴 𝐶 · 𝐷))
 
Theoremfprodcl2lem 11644* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.) (Revised by Jim Kingdon, 17-Aug-2024.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑𝐴 ≠ ∅)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)
 
Theoremfprodcllem 11645* Finite product closure lemma. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 1 ∈ 𝑆)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)
 
Theoremfprodcl 11646* Closure of a finite product of complex numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℂ)
 
Theoremfprodrecl 11647* Closure of a finite product of real numbers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ)
 
Theoremfprodzcl 11648* Closure of a finite product of integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℤ)
 
Theoremfprodnncl 11649* Closure of a finite product of positive integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℕ)
 
Theoremfprodrpcl 11650* Closure of a finite product of positive reals. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ+)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ+)
 
Theoremfprodnn0cl 11651* Closure of a finite product of nonnegative integers. (Contributed by Scott Fenton, 14-Dec-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℕ0)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℕ0)
 
Theoremfprodcllemf 11652* Finite product closure lemma. A version of fprodcllem 11645 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵𝑆)    &   (𝜑 → 1 ∈ 𝑆)       (𝜑 → ∏𝑘𝐴 𝐵𝑆)
 
Theoremfprodreclf 11653* Closure of a finite product of real numbers. A version of fprodrecl 11647 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℝ)
 
Theoremfprodfac 11654* Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)
 
Theoremfprodabs 11655* The absolute value of a finite product. (Contributed by Scott Fenton, 25-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)       (𝜑 → (abs‘∏𝑘 ∈ (𝑀...𝑁)𝐴) = ∏𝑘 ∈ (𝑀...𝑁)(abs‘𝐴))
 
Theoremfprodeq0 11656* Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑁𝑍)    &   ((𝜑𝑘𝑍) → 𝐴 ∈ ℂ)    &   ((𝜑𝑘 = 𝑁) → 𝐴 = 0)       ((𝜑𝐾 ∈ (ℤ𝑁)) → ∏𝑘 ∈ (𝑀...𝐾)𝐴 = 0)
 
Theoremfprodshft 11657* Shift the index of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝑘𝐾) → 𝐴 = 𝐵)       (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝑀 + 𝐾)...(𝑁 + 𝐾))𝐵)
 
Theoremfprodrev 11658* Reversal of a finite product. (Contributed by Scott Fenton, 5-Jan-2018.)
(𝜑𝐾 ∈ ℤ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑗 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)    &   (𝑗 = (𝐾𝑘) → 𝐴 = 𝐵)       (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)𝐴 = ∏𝑘 ∈ ((𝐾𝑁)...(𝐾𝑀))𝐵)
 
Theoremfprodconst 11659* The product of constant terms (𝑘 is not free in 𝐵). (Contributed by Scott Fenton, 12-Jan-2018.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ ℂ) → ∏𝑘𝐴 𝐵 = (𝐵↑(♯‘𝐴)))
 
Theoremfprodap0 11660* A finite product of nonzero terms is nonzero. (Contributed by Scott Fenton, 15-Jan-2018.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 # 0)       (𝜑 → ∏𝑘𝐴 𝐵 # 0)
 
Theoremfprod2dlemstep 11661* Lemma for fprod2d 11662- induction step. (Contributed by Scott Fenton, 30-Jan-2018.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)    &   (𝜑 → ¬ 𝑦𝑥)    &   (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴)    &   (𝜑𝑥 ∈ Fin)    &   (𝜓 ↔ ∏𝑗𝑥𝑘𝐵 𝐶 = ∏𝑧 𝑗𝑥 ({𝑗} × 𝐵)𝐷)       ((𝜑𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘𝐵 𝐶 = ∏𝑧 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷)
 
Theoremfprod2d 11662* Write a double product as a product over a two-dimensional region. Compare fsum2d 11474. (Contributed by Scott Fenton, 30-Jan-2018.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 𝑗𝐴 ({𝑗} × 𝐵)𝐷)
 
Theoremfprodxp 11663* Combine two products into a single product over the cartesian product. (Contributed by Scott Fenton, 1-Feb-2018.)
(𝑧 = ⟨𝑗, 𝑘⟩ → 𝐷 = 𝐶)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑧 ∈ (𝐴 × 𝐵)𝐷)
 
Theoremfprodcnv 11664* Transform a product region using the converse operation. (Contributed by Scott Fenton, 1-Feb-2018.)
(𝑥 = ⟨𝑗, 𝑘⟩ → 𝐵 = 𝐷)    &   (𝑦 = ⟨𝑘, 𝑗⟩ → 𝐶 = 𝐷)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → Rel 𝐴)    &   ((𝜑𝑥𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑥𝐴 𝐵 = ∏𝑦 𝐴𝐶)
 
Theoremfprodcom2fi 11665* Interchange order of multiplication. Note that 𝐵(𝑗) and 𝐷(𝑘) are not necessarily constant expressions. (Contributed by Scott Fenton, 1-Feb-2018.) (Proof shortened by JJ, 2-Aug-2021.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑𝑗𝐴) → 𝐵 ∈ Fin)    &   ((𝜑𝑘𝐶) → 𝐷 ∈ Fin)    &   (𝜑 → ((𝑗𝐴𝑘𝐵) ↔ (𝑘𝐶𝑗𝐷)))    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐸 ∈ ℂ)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐸 = ∏𝑘𝐶𝑗𝐷 𝐸)
 
Theoremfprodcom 11666* Interchange product order. (Contributed by Scott Fenton, 2-Feb-2018.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐵)) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑗𝐴𝑘𝐵 𝐶 = ∏𝑘𝐵𝑗𝐴 𝐶)
 
Theoremfprod0diagfz 11667* Two ways to express "the product of 𝐴(𝑗, 𝑘) over the triangular region 𝑀𝑗, 𝑀𝑘, 𝑗 + 𝑘𝑁. Compare fisum0diag 11480. (Contributed by Scott Fenton, 2-Feb-2018.)
((𝜑 ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗)))) → 𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → ∏𝑗 ∈ (0...𝑁)∏𝑘 ∈ (0...(𝑁𝑗))𝐴 = ∏𝑘 ∈ (0...𝑁)∏𝑗 ∈ (0...(𝑁𝑘))𝐴)
 
Theoremfprodrec 11668* The finite product of reciprocals is the reciprocal of the product. (Contributed by Jim Kingdon, 28-Aug-2024.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 # 0)       (𝜑 → ∏𝑘𝐴 (1 / 𝐵) = (1 / ∏𝑘𝐴 𝐵))
 
Theoremfproddivap 11669* The quotient of two finite products. (Contributed by Scott Fenton, 15-Jan-2018.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 # 0)       (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
 
Theoremfproddivapf 11670* The quotient of two finite products. A version of fproddivap 11669 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐶 # 0)       (𝜑 → ∏𝑘𝐴 (𝐵 / 𝐶) = (∏𝑘𝐴 𝐵 / ∏𝑘𝐴 𝐶))
 
Theoremfprodsplitf 11671* Split a finite product into two parts. A version of fprodsplit 11636 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑈 = (𝐴𝐵))    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑𝑘𝑈) → 𝐶 ∈ ℂ)       (𝜑 → ∏𝑘𝑈 𝐶 = (∏𝑘𝐴 𝐶 · ∏𝑘𝐵 𝐶))
 
Theoremfprodsplitsn 11672* Separate out a term in a finite product. See also fprodunsn 11643 which is the same but with a distinct variable condition in place of 𝑘𝜑. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   𝑘𝐷    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑉)    &   (𝜑 → ¬ 𝐵𝐴)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)    &   (𝑘 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐷 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (𝐴 ∪ {𝐵})𝐶 = (∏𝑘𝐴 𝐶 · 𝐷))
 
Theoremfprodsplit1f 11673* Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝑘𝐷)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑘 = 𝐶) → 𝐵 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
 
Theoremfprodclf 11674* Closure of a finite product of complex numbers. A version of fprodcl 11646 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → ∏𝑘𝐴 𝐵 ∈ ℂ)
 
Theoremfprodap0f 11675* A finite product of terms apart from zero is apart from zero. A version of fprodap0 11660 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Revised by Jim Kingdon, 30-Aug-2024.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   ((𝜑𝑘𝐴) → 𝐵 # 0)       (𝜑 → ∏𝑘𝐴 𝐵 # 0)
 
Theoremfprodge0 11676* If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)       (𝜑 → 0 ≤ ∏𝑘𝐴 𝐵)
 
Theoremfprodeq0g 11677* Any finite product containing a zero term is itself zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)    &   (𝜑𝐶𝐴)    &   ((𝜑𝑘 = 𝐶) → 𝐵 = 0)       (𝜑 → ∏𝑘𝐴 𝐵 = 0)
 
Theoremfprodge1 11678* 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.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 1 ≤ 𝐵)       (𝜑 → 1 ≤ ∏𝑘𝐴 𝐵)
 
Theoremfprodle 11679* If all the terms of two finite products are nonnegative and compare, so do the two products. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 0 ≤ 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℝ)    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → ∏𝑘𝐴 𝐵 ≤ ∏𝑘𝐴 𝐶)
 
Theoremfprodmodd 11680* If all factors of two finite products are equal modulo 𝑀, the products are equal modulo 𝑀. (Contributed by AV, 7-Jul-2021.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℤ)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℤ)    &   (𝜑𝑀 ∈ ℕ)    &   ((𝜑𝑘𝐴) → (𝐵 mod 𝑀) = (𝐶 mod 𝑀))       (𝜑 → (∏𝑘𝐴 𝐵 mod 𝑀) = (∏𝑘𝐴 𝐶 mod 𝑀))
 
4.9  Elementary trigonometry
 
4.9.1  The exponential, sine, and cosine functions
 
Syntaxce 11681 Extend class notation to include the exponential function.
class exp
 
Syntaxceu 11682 Extend class notation to include Euler's constant e = 2.71828....
class e
 
Syntaxcsin 11683 Extend class notation to include the sine function.
class sin
 
Syntaxccos 11684 Extend class notation to include the cosine function.
class cos
 
Syntaxctan 11685 Extend class notation to include the tangent function.
class tan
 
Syntaxcpi 11686 Extend class notation to include the constant pi, π = 3.14159....
class π
 
Definitiondf-ef 11687* Define the exponential function. Its value at the complex number 𝐴 is (exp‘𝐴) and is called the "exponential of 𝐴"; see efval 11700. (Contributed by NM, 14-Mar-2005.)
exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
 
Definitiondf-e 11688 Define Euler's constant e = 2.71828.... (Contributed by NM, 14-Mar-2005.)
e = (exp‘1)
 
Definitiondf-sin 11689 Define the sine function. (Contributed by NM, 14-Mar-2005.)
sin = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) − (exp‘(-i · 𝑥))) / (2 · i)))
 
Definitiondf-cos 11690 Define the cosine function. (Contributed by NM, 14-Mar-2005.)
cos = (𝑥 ∈ ℂ ↦ (((exp‘(i · 𝑥)) + (exp‘(-i · 𝑥))) / 2))
 
Definitiondf-tan 11691 Define the tangent function. We define it this way for cmpt 4079, which requires the form (𝑥𝐴𝐵). (Contributed by Mario Carneiro, 14-Mar-2014.)
tan = (𝑥 ∈ (cos “ (ℂ ∖ {0})) ↦ ((sin‘𝑥) / (cos‘𝑥)))
 
Definitiondf-pi 11692 Define the constant pi, π = 3.14159..., which is the smallest positive number whose sine is zero. Definition of π in [Gleason] p. 311. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by AV, 14-Sep-2020.)
π = inf((ℝ+ ∩ (sin “ {0})), ℝ, < )
 
Theoremeftcl 11693 Closure of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 11-Sep-2007.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℂ)
 
Theoremreeftcl 11694 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 ∈ ℕ0) → ((𝐴𝐾) / (!‘𝐾)) ∈ ℝ)
 
Theoremeftabs 11695 The absolute value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 23-Nov-2007.)
((𝐴 ∈ ℂ ∧ 𝐾 ∈ ℕ0) → (abs‘((𝐴𝐾) / (!‘𝐾))) = (((abs‘𝐴)↑𝐾) / (!‘𝐾)))
 
Theoremeftvalcn 11696* 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 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
 
Theoremefcllemp 11697* Lemma for efcl 11703. The series that defines the exponential function converges. The ratio test cvgratgt0 11572 is used to show convergence. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑 → (2 · (abs‘𝐴)) < 𝐾)       (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
 
Theoremefcllem 11698* Lemma for efcl 11703. The series that defines the exponential function converges. (Contributed by NM, 26-Apr-2005.) (Revised by Jim Kingdon, 8-Dec-2022.)
𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
 
Theoremef0lem 11699* 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 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))       (𝐴 = 0 → seq0( + , 𝐹) ⇝ 1)
 
Theoremefval 11700* Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
(𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))
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