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Type | Label | Description |
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Statement | ||
Theorem | iseqf1olemqval 10101* | Lemma for seq3f1o 10118. 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 10102* | Lemma for seq3f1o 10118. (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 10103* | Lemma for seq3f1o 10118. (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 10104* | Lemma for seq3f1o 10118. (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 10105* | Lemma for seq3f1o 10118. 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 10106* | Lemma for seq3f1o 10118. 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 10107* | Lemma for seq3f1o 10118. 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 10108* | Lemma for seq3f1o 10118. 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 10109* | Lemma for seq3f1o 10118. 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 10110* | Lemma for seq3f1o 10118. 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 10111* | Lemma for seq3f1o 10118. (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 10112* | Lemma for seq3f1o 10118. 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 10113* | Lemma for seq3f1o 10118. 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 10114* | Lemma for seq3f1o 10118. 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 10115* | Lemma for seq3f1o 10118. 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 10116* | Lemma for seq3f1o 10118. 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 10117* | Lemma for seq3f1o 10118. 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 10118* | 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 10119* | 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 10120* | 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 10121* | 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 10122* | 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 10123* | 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 10124* | 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 10125* | 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 10126* | 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 10127* | 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 10128 | 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 10129 | 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 10130* | 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 10131* | 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 10132* | 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 10133 | Extend class notation to include exponentiation of a complex number to an integer power. |
class ↑ | ||
Definition | df-exp 10134* |
Define exponentiation to nonnegative integer powers. For example,
(5↑2) = 25 (ex-exp 12542).
This definition is not meant to be used directly; instead, exp0 10138 and expp1 10141 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 10139). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 12542). 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 10135 | Lemma for exp3val 10136. 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 10136 | 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 10137 | 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 10138 | 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 10139 | 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 10140 | 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 10141 | 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 10142 | 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 10143 | 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 10144 | 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 10145* | 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 10146* | 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 10147 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcl 10148 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | zexpcl 10149 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | ||
Theorem | qexpcl 10150 | Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | reexpcl 10151 | Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expcl 10152 | Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | rpexpcl 10153 | 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 10154 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | qexpclz 10155 | Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | m1expcl2 10156 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | ||
Theorem | m1expcl 10157 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ) | ||
Theorem | expclzaplem 10158* | Closure law for integer exponentiation. Lemma for expclzap 10159 and expap0i 10166. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ {𝑧 ∈ ℂ ∣ 𝑧 # 0}) | ||
Theorem | expclzap 10159 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | nn0expcli 10160 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝐴↑𝑁) ∈ ℕ0 | ||
Theorem | nn0sqcl 10161 | 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 10162 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) | ||
Theorem | 1exp 10163 | 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 10164 | 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 10165 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 10165 | 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 10166 | 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 10167 | 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 10168 | 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 10169 | Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | ||
Theorem | expge0 10170 | 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 10171 | 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 10172 | 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 10173 | 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 10174 | 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 10175 | 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 10176 | 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 10177 | Lemma for expaddzap 10178. (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 10178 | 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 10179 | 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 10180 | 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 10181 | Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) | ||
Theorem | expsubap 10182 | 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 10183 | 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 10184 | 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 10185 | 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 10186 | 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)) | ||
Theorem | leexp2a 10187 | Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) | ||
Theorem | leexp2r 10188 | Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)) | ||
Theorem | leexp1a 10189 | Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴↑𝑁) ≤ (𝐵↑𝑁)) | ||
Theorem | exple1 10190 | Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ 1) ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ≤ 1) | ||
Theorem | expubnd 10191 | An upper bound on 𝐴↑𝑁 when 2 ≤ 𝐴. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 2 ≤ 𝐴) → (𝐴↑𝑁) ≤ ((2↑𝑁) · ((𝐴 − 1)↑𝑁))) | ||
Theorem | sqval 10192 | Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | ||
Theorem | sqneg 10193 | The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | ||
Theorem | sqsubswap 10194 | Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵)↑2) = ((𝐵 − 𝐴)↑2)) | ||
Theorem | sqcl 10195 | Closure of square. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | ||
Theorem | sqmul 10196 | Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | ||
Theorem | sqeq0 10197 | A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0)) | ||
Theorem | sqdivap 10198 | Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | sqne0 10199 | A number is nonzero iff its square is nonzero. See also sqap0 10200 which is the same but with not equal changed to apart. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0)) | ||
Theorem | sqap0 10200 | A number is apart from zero iff its square is apart from zero. (Contributed by Jim Kingdon, 13-Aug-2021.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) # 0 ↔ 𝐴 # 0)) |
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