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Type | Label | Description |
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Statement | ||
Theorem | seq3shft2 10201* | Shifting the index set of a sequence. (Contributed by Jim Kingdon, 15-Aug-2021.) (Revised by Jim Kingdon, 7-Apr-2023.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝐾))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 𝐾))) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq(𝑀 + 𝐾)( + , 𝐺)‘(𝑁 + 𝐾))) | ||
Theorem | serf 10202* | An infinite series of complex terms is a function from ℕ to ℂ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) | ||
Theorem | serfre 10203* | An infinite series of real numbers is a function from ℕ to ℝ. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ) | ||
Theorem | monoord 10204* | Ordering relation for a monotonic sequence, increasing case. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) ⇒ ⊢ (𝜑 → (𝐹‘𝑀) ≤ (𝐹‘𝑁)) | ||
Theorem | monoord2 10205* | Ordering relation for a monotonic sequence, decreasing case. (Contributed by Mario Carneiro, 18-Jul-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) | ||
Theorem | ser3mono 10206* | The partial sums in an infinite series of positive terms form a monotonic sequence. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → 0 ≤ (𝐹‘𝑥)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ≤ (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | seq3split 10207* | Split a sequence into two sequences. (Contributed by Jim Kingdon, 16-Aug-2021.) (Revised by Jim Kingdon, 21-Oct-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) & ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐾)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝑁) = ((seq𝐾( + , 𝐹)‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))) | ||
Theorem | seq3-1p 10208* | Removing the first term from a sequence. (Contributed by Jim Kingdon, 16-Aug-2021.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = ((𝐹‘𝑀) + (seq(𝑀 + 1)( + , 𝐹)‘𝑁))) | ||
Theorem | seq3caopr3 10209* | Lemma for seq3caopr2 10210. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by Jim Kingdon, 22-Apr-2023.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) & ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (((seq𝑀( + , 𝐹)‘𝑛)𝑄(seq𝑀( + , 𝐺)‘𝑛)) + ((𝐹‘(𝑛 + 1))𝑄(𝐺‘(𝑛 + 1)))) = (((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))𝑄((seq𝑀( + , 𝐺)‘𝑛) + (𝐺‘(𝑛 + 1))))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | seq3caopr2 10210* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by Mario Carneiro, 30-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))) → ((𝑥𝑄𝑧) + (𝑦𝑄𝑤)) = ((𝑥 + 𝑦)𝑄(𝑧 + 𝑤))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘)𝑄(𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁)𝑄(seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | seq3caopr 10211* | The sum of two infinite series (generalized to an arbitrary commutative and associative operation). (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | iseqf1olemkle 10212* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 21-Aug-2022.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) ⇒ ⊢ (𝜑 → 𝐾 ≤ (◡𝐽‘𝐾)) | ||
Theorem | iseqf1olemklt 10213* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 21-Aug-2022.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) ⇒ ⊢ (𝜑 → 𝐾 < (◡𝐽‘𝐾)) | ||
Theorem | iseqf1olemqcl 10214 | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 27-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) ⇒ ⊢ (𝜑 → if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴)) ∈ (𝑀...𝑁)) | ||
Theorem | iseqf1olemqval 10215* | Lemma for seq3f1o 10232. Value of the function 𝑄. (Contributed by Jim Kingdon, 28-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ⇒ ⊢ (𝜑 → (𝑄‘𝐴) = if(𝐴 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝐴 = 𝐾, 𝐾, (𝐽‘(𝐴 − 1))), (𝐽‘𝐴))) | ||
Theorem | iseqf1olemnab 10216* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 27-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ⇒ ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐾...(◡𝐽‘𝐾)) ∧ ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾)))) | ||
Theorem | iseqf1olemab 10217* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 27-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ (𝜑 → 𝐴 ∈ (𝐾...(◡𝐽‘𝐾))) & ⊢ (𝜑 → 𝐵 ∈ (𝐾...(◡𝐽‘𝐾))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | iseqf1olemnanb 10218* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 27-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ (𝜑 → ¬ 𝐴 ∈ (𝐾...(◡𝐽‘𝐾))) & ⊢ (𝜑 → ¬ 𝐵 ∈ (𝐾...(◡𝐽‘𝐾))) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | iseqf1olemqf 10219* | Lemma for seq3f1o 10232. Domain and codomain of 𝑄. (Contributed by Jim Kingdon, 26-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ⇒ ⊢ (𝜑 → 𝑄:(𝑀...𝑁)⟶(𝑀...𝑁)) | ||
Theorem | iseqf1olemmo 10220* | Lemma for seq3f1o 10232. Showing that 𝑄 is one-to-one. (Contributed by Jim Kingdon, 27-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐵 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → (𝑄‘𝐴) = (𝑄‘𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | iseqf1olemqf1o 10221* | Lemma for seq3f1o 10232. 𝑄 is a permutation of (𝑀...𝑁). 𝑄 is formed from the constant portion of 𝐽, followed by the single element 𝐾 (at position 𝐾), followed by the rest of J (with the 𝐾 deleted and the elements before 𝐾 moved one position later to fill the gap). (Contributed by Jim Kingdon, 21-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ⇒ ⊢ (𝜑 → 𝑄:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | ||
Theorem | iseqf1olemqk 10222* | Lemma for seq3f1o 10232. 𝑄 is constant for one more position than 𝐽 is. (Contributed by Jim Kingdon, 21-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ (𝑀...𝐾)(𝑄‘𝑥) = 𝑥) | ||
Theorem | iseqf1olemjpcl 10223* | Lemma for seq3f1o 10232. A closure lemma involving 𝐽 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (⦋𝐽 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆) | ||
Theorem | iseqf1olemqpcl 10224* | Lemma for seq3f1o 10232. A closure lemma involving 𝑄 and 𝑃. (Contributed by Jim Kingdon, 29-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (⦋𝑄 / 𝑓⦌𝑃‘𝑥) ∈ 𝑆) | ||
Theorem | iseqf1olemfvp 10225* | Lemma for seq3f1o 10232. (Contributed by Jim Kingdon, 30-Aug-2022.) |
⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝑇:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → (⦋𝑇 / 𝑓⦌𝑃‘𝐴) = (𝐺‘(𝑇‘𝐴))) | ||
Theorem | seq3f1olemqsumkj 10226* | Lemma for seq3f1o 10232. 𝑄 gives the same sum as 𝐽 in the range (𝐾...(◡𝐽‘𝐾)). (Contributed by Jim Kingdon, 29-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘(◡𝐽‘𝐾)) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘(◡𝐽‘𝐾))) | ||
Theorem | seq3f1olemqsumk 10227* | Lemma for seq3f1o 10232. 𝑄 gives the same sum as 𝐽 in the range (𝐾...𝑁). (Contributed by Jim Kingdon, 22-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → (seq𝐾( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝐾( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)) | ||
Theorem | seq3f1olemqsum 10228* | Lemma for seq3f1o 10232. 𝑄 gives the same sum as 𝐽. (Contributed by Jim Kingdon, 21-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐾 ≠ (◡𝐽‘𝐾)) & ⊢ 𝑄 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑄 / 𝑓⦌𝑃)‘𝑁)) | ||
Theorem | seq3f1olemstep 10229* | Lemma for seq3f1o 10232. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥) & ⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) | ||
Theorem | seq3f1olemp 10230* | Lemma for seq3f1o 10232. Existence of a constant permutation of (𝑀...𝑁) which leads to the same sum as the permutation 𝐹 itself. (Contributed by Jim Kingdon, 18-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ 𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀))) & ⊢ 𝑃 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) | ||
Theorem | seq3f1oleml 10231* | Lemma for seq3f1o 10232. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ 𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) | ||
Theorem | seq3f1o 10232* | Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) | ||
Theorem | ser3add 10233* | The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 4-Oct-2022.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | ser3sub 10234* | The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Jim Kingdon, 22-Apr-2023.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | seq3id3 10235* | 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.) (Revised by Jim Kingdon, 8-Apr-2023.) |
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) | ||
Theorem | seq3id 10236* | 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 Jim Kingdon, 8-Apr-2023.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑥) & ⊢ (𝜑 → 𝑍 ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → (𝐹‘𝑁) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑥) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) | ||
Theorem | seq3id2 10237* | 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 Jim Kingdon, 12-Nov-2022.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥) & ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾)) & ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | seq3homo 10238* | Apply a homomorphism to a sequence. (Contributed by Jim Kingdon, 10-Oct-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑄𝑦) ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)) | ||
Theorem | seq3z 10239* | 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.) (Revised by Jim Kingdon, 23-Apr-2023.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) | ||
Theorem | seqfeq3 10240* | 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.) |
⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) | ||
Theorem | seq3distr 10241* | The distributive property for series. (Contributed by Jim Kingdon, 10-Oct-2022.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑇𝑦) ∈ 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | ser0 10242 | 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 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (seq𝑀( + , (𝑍 × {0}))‘𝑁) = 0) | ||
Theorem | ser0f 10243 | A zero-valued infinite series is equal to the constant zero function. (Contributed by Mario Carneiro, 8-Feb-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → seq𝑀( + , (𝑍 × {0})) = (𝑍 × {0})) | ||
Theorem | fser0const 10244* | Simplifying an expression which turns out just to be a constant zero sequence. (Contributed by Jim Kingdon, 16-Sep-2022.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ if(𝑛 ≤ 𝑁, ((𝑍 × {0})‘𝑛), 0)) = (𝑍 × {0})) | ||
Theorem | ser3ge0 10245* | A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | ser3le 10246* | Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁)) | ||
Syntax | cexp 10247 | Extend class notation to include exponentiation of a complex number to an integer power. |
class ↑ | ||
Definition | df-exp 10248* |
Define exponentiation to nonnegative integer powers. For example,
(5↑2) = 25 (ex-exp 12835).
This definition is not meant to be used directly; instead, exp0 10252 and expp1 10255 provide the standard recursive definition. The up-arrow notation is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976) and is convenient for us since we don't have superscripts. 10-Jun-2005: The definition was extended to include zero exponents, so that 0↑0 = 1 per the convention of Definition 10-4.1 of [Gleason] p. 134 (0exp0e1 10253). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 12835). The case 𝑥 = 0, 𝑦 < 0 gives the value (1 / 0), so we will avoid this case in our theorems. (Contributed by Raph Levien, 20-May-2004.) (Revised by NM, 15-Oct-2004.) |
⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))))) | ||
Theorem | exp3vallem 10249 | Lemma for exp3val 10250. If we take a complex number apart from zero and raise it to a positive integer power, the result is apart from zero. (Contributed by Jim Kingdon, 7-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (seq1( · , (ℕ × {𝐴}))‘𝑁) # 0) | ||
Theorem | exp3val 10250 | Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))))) | ||
Theorem | expnnval 10251 | Value of exponentiation to positive integer powers. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (seq1( · , (ℕ × {𝐴}))‘𝑁)) | ||
Theorem | exp0 10252 | Value of a complex number raised to the 0th power. Note that under our definition, 0↑0 = 1, following the convention used by Gleason. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑0) = 1) | ||
Theorem | 0exp0e1 10253 | 0↑0 = 1 (common case). This is our convention. It follows the convention used by Gleason; see Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0↑0) = 1 | ||
Theorem | exp1 10254 | Value of a complex number raised to the first power. (Contributed by NM, 20-Oct-2004.) (Revised by Mario Carneiro, 2-Jul-2013.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | ||
Theorem | expp1 10255 | Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. (Contributed by NM, 20-May-2005.) (Revised by Mario Carneiro, 2-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expnegap0 10256 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expineg2 10257 | Value of a complex number raised to a negative integer power. (Contributed by Jim Kingdon, 8-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0)) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) | ||
Theorem | expn1ap0 10258 | A number to the negative one power is the reciprocal. (Contributed by Jim Kingdon, 8-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝐴↑-1) = (1 / 𝐴)) | ||
Theorem | expcllem 10259* | Lemma for proving nonnegative integer exponentiation closure laws. (Contributed by NM, 14-Dec-2005.) |
⊢ 𝐹 ⊆ ℂ & ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) & ⊢ 1 ∈ 𝐹 ⇒ ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ ℕ0) → (𝐴↑𝐵) ∈ 𝐹) | ||
Theorem | expcl2lemap 10260* | Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.) |
⊢ 𝐹 ⊆ ℂ & ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 · 𝑦) ∈ 𝐹) & ⊢ 1 ∈ 𝐹 & ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑥 # 0) → (1 / 𝑥) ∈ 𝐹) ⇒ ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℤ) → (𝐴↑𝐵) ∈ 𝐹) | ||
Theorem | nnexpcl 10261 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcl 10262 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | zexpcl 10263 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | ||
Theorem | qexpcl 10264 | Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | reexpcl 10265 | Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expcl 10266 | Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | rpexpcl 10267 | Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ+) | ||
Theorem | reexpclzap 10268 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | qexpclz 10269 | Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | m1expcl2 10270 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | ||
Theorem | m1expcl 10271 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ) | ||
Theorem | expclzaplem 10272* | Closure law for integer exponentiation. Lemma for expclzap 10273 and expap0i 10280. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) | ||
Theorem | expclzap 10273 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | nn0expcli 10274 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝐴↑𝑁) ∈ ℕ0 | ||
Theorem | nn0sqcl 10275 | The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.) |
⊢ (𝐴 ∈ ℕ0 → (𝐴↑2) ∈ ℕ0) | ||
Theorem | expm1t 10276 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) | ||
Theorem | 1exp 10277 | Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) | ||
Theorem | expap0 10278 | Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 10279 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." (Remark of [Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # 0 ↔ 𝐴 # 0)) | ||
Theorem | expeq0 10279 | Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0)) | ||
Theorem | expap0i 10280 | Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) # 0) | ||
Theorem | expgt0 10281 | Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 0 < 𝐴) → 0 < (𝐴↑𝑁)) | ||
Theorem | expnegzap 10282 | Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | 0exp 10283 | Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | ||
Theorem | expge0 10284 | Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑁)) | ||
Theorem | expge1 10285 | Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 1 ≤ 𝐴) → 1 ≤ (𝐴↑𝑁)) | ||
Theorem | expgt1 10286 | Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ ∧ 1 < 𝐴) → 1 < (𝐴↑𝑁)) | ||
Theorem | mulexp 10287 | Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 13-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁))) | ||
Theorem | mulexpzap 10288 | Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁))) | ||
Theorem | exprecap 10289 | Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expadd 10290 | Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expaddzaplem 10291 | Lemma for expaddzap 10292. (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expaddzap 10292 | Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expmul 10293 | Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁)) | ||
Theorem | expmulzap 10294 | Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁)) | ||
Theorem | m1expeven 10295 | Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) | ||
Theorem | expsubap 10296 | Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁))) | ||
Theorem | expp1zap 10297 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expm1ap 10298 | Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴)) | ||
Theorem | expdivap 10299 | Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁))) | ||
Theorem | ltexp2a 10300 | Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 < 𝐴 ∧ 𝑀 < 𝑁)) → (𝐴↑𝑀) < (𝐴↑𝑁)) |
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