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Type | Label | Description |
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Statement | ||
Theorem | seqf1olem2a 12701* | Lemma for seqf1o 12704. (Contributed by Mario Carneiro, 24-Apr-2016.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐶 ⊆ 𝑆) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐴) ⇒ ⊢ (𝜑 → ((𝐺‘𝐾) + (seq𝑀( + , 𝐺)‘𝑁)) = ((seq𝑀( + , 𝐺)‘𝑁) + (𝐺‘𝐾))) | ||
Theorem | seqf1olem1 12702* | Lemma for seqf1o 12704. (Contributed by Mario Carneiro, 26-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐶 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) & ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶) & ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) & ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1)) ⇒ ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) | ||
Theorem | seqf1olem2 12703* | Lemma for seqf1o 12704. (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐶 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) & ⊢ (𝜑 → 𝐺:(𝑀...(𝑁 + 1))⟶𝐶) & ⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) & ⊢ 𝐾 = (◡𝐹‘(𝑁 + 1)) & ⊢ (𝜑 → ∀𝑔∀𝑓((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝑔:(𝑀...𝑁)⟶𝐶) → (seq𝑀( + , (𝑔 ∘ 𝑓))‘𝑁) = (seq𝑀( + , 𝑔)‘𝑁))) ⇒ ⊢ (𝜑 → (seq𝑀( + , (𝐺 ∘ 𝐹))‘(𝑁 + 1)) = (seq𝑀( + , 𝐺)‘(𝑁 + 1))) | ||
Theorem | seqf1o 12704* | Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐶 ⊆ 𝑆) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁)) | ||
Theorem | seradd 12705* | The sum of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 26-May-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) + (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | sersub 12706* | The difference of two infinite series. (Contributed by NM, 17-Mar-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℂ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = ((seq𝑀( + , 𝐹)‘𝑁) − (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | seqid3 12707* | A sequence that consists entirely of zeroes (or whatever the identity 𝑍 is for operation +) sums to zero. (Contributed by Mario Carneiro, 15-Dec-2014.) |
⊢ (𝜑 → (𝑍 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = 𝑍) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) | ||
Theorem | seqid 12708* | Discard the first few terms of a sequence that starts with all zeroes (or whatever the identity 𝑍 is for operation +). (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ ((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 → (seq𝑀( + , 𝐹) ↾ (ℤ≥‘𝑁)) = seq𝑁( + , 𝐹)) | ||
Theorem | seqid2 12709* | The last few terms of a sequence that ends with all zeroes (or whatever the identity 𝑍 is for operation +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ ((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 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | seqhomo 12710* | Apply a homomorphism to a sequence. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐻‘(𝑥 + 𝑦)) = ((𝐻‘𝑥)𝑄(𝐻‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐻‘(𝐹‘𝑥)) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (𝐻‘(seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀(𝑄, 𝐺)‘𝑁)) | ||
Theorem | seqz 12711* | 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.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍) & ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) & ⊢ (𝜑 → 𝑁 ∈ 𝑉) & ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍) | ||
Theorem | seqfeq4 12712* | Equality of series under different addition operations which agree on an additively closed subset. (Contributed by Mario Carneiro, 25-Apr-2016.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑥𝑄𝑦)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀(𝑄, 𝐹)‘𝑁)) | ||
Theorem | seqfeq3 12713* | 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 | seqdistr 12714* | The distributive property for series. (Contributed by Mario Carneiro, 28-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐶𝑇(𝑥 + 𝑦)) = ((𝐶𝑇𝑥) + (𝐶𝑇𝑦))) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐺‘𝑥) ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝐶𝑇(𝐺‘𝑥))) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (𝐶𝑇(seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | ser0 12715 | 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 12716 | 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 | serge0 12717* | 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 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 0 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 0 ≤ (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | serle 12718* | Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 27-May-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≤ (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ≤ (seq𝑀( + , 𝐺)‘𝑁)) | ||
Theorem | ser1const 12719 | Value of the partial series sum of a constant function. (Contributed by NM, 8-Aug-2005.) (Revised by Mario Carneiro, 16-Feb-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (seq1( + , (ℕ × {𝐴}))‘𝑁) = (𝑁 · 𝐴)) | ||
Theorem | seqof 12720* | Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))) ⇒ ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁))) | ||
Theorem | seqof2 12721* | Distribute function operation through a sequence. Maps-to notation version of seqof 12720. (Contributed by Mario Carneiro, 7-Jul-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → (𝑀...𝑁) ⊆ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) → 𝑋 ∈ 𝑊) ⇒ ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , (𝑥 ∈ 𝐵 ↦ (𝑧 ∈ 𝐴 ↦ 𝑋)))‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , (𝑥 ∈ 𝐵 ↦ 𝑋))‘𝑁))) | ||
Syntax | cexp 12722 | Extend class notation to include exponentiation of a complex number to an integer power. |
class ↑ | ||
Definition | df-exp 12723* |
Define exponentiation to nonnegative integer powers. For example,
(5↑2) = 25 (ex-exp 26699).
This definition is not meant to be used directly; instead, exp0 12726 and expp1 12729 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 12727). 4-Jun-2014: The definition was extended to include negative integer exponents. For example, (-3↑-2) = (1 / 9) (ex-exp 26699). 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 | expval 12724 | Value of exponentiation to integer powers. (Contributed by NM, 20-May-2004.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))))) | ||
Theorem | expnnval 12725 | 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 12726 | 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 12727 | 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 12728 | 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 12729 | 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 | expneg 12730 | Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expneg2 12731 | Value of a complex number raised to a negative integer power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) | ||
Theorem | expn1 12732 | A number to the negative one power is the reciprocal. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑-1) = (1 / 𝐴)) | ||
Theorem | expcllem 12733* | 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 | expcl2lem 12734* | Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.) |
⊢ 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 12735 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcl 12736 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | zexpcl 12737 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | ||
Theorem | qexpcl 12738 | Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | reexpcl 12739 | Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expcl 12740 | Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | rpexpcl 12741 | 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 | reexpclz 12742 | Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | qexpclz 12743 | Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | m1expcl2 12744 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | ||
Theorem | m1expcl 12745 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ) | ||
Theorem | expclzlem 12746 | Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0})) | ||
Theorem | expclz 12747 | Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | nn0expcli 12748 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝐴↑𝑁) ∈ ℕ0 | ||
Theorem | nn0sqcl 12749 | 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 12750 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) | ||
Theorem | 1exp 12751 | 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 | expeq0 12752 | 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 | expne0 12753 | Positive integer exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) ≠ 0 ↔ 𝐴 ≠ 0)) | ||
Theorem | expne0i 12754 | Nonnegative integer exponentiation is nonzero if its mantissa is nonzero. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ≠ 0) | ||
Theorem | expgt0 12755 | 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 | expnegz 12756 | 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 12757 | Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | ||
Theorem | expge0 12758 | 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 12759 | 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 12760 | 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 12761 | 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 | mulexpz 12762 | Integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝑁 ∈ ℤ) → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁))) | ||
Theorem | exprec 12763 | Nonnegative integer exponentiation of a reciprocal. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expadd 12764 | 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 | expaddzlem 12765 | Lemma for expaddz 12766. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℝ ∧ -𝑀 ∈ ℕ) ∧ 𝑁 ∈ ℕ0) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expaddz 12766 | Sum of exponents law for integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expmul 12767 | 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 | expmulz 12768 | Product of exponents law for integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 7-Jul-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁)) | ||
Theorem | m1expeven 12769 | Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) | ||
Theorem | expsub 12770 | Exponent subtraction law for nonnegative integer exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁))) | ||
Theorem | expp1z 12771 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expm1 12772 | Value of a complex number raised to an integer power minus one. (Contributed by NM, 25-Dec-2008.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴)) | ||
Theorem | expdiv 12773 | Nonnegative integer exponentiation of a quotient. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝑁 ∈ ℕ0) → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁))) | ||
Theorem | ltexp2a 12774 | 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 | expcan 12775 | Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ 𝑀 = 𝑁)) | ||
Theorem | ltexp2 12776 | Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) | ||
Theorem | leexp2 12777 | Ordering law for exponentiation. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁))) | ||
Theorem | leexp2a 12778 | 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 | ltexp2r 12779 | The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ (((𝐴 ∈ ℝ+ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀))) | ||
Theorem | leexp2r 12780 | 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 12781 | 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 12782 | 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 12783 | 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 12784 | Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | ||
Theorem | sqneg 12785 | The square of the negative of a number. (Contributed by NM, 15-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | ||
Theorem | sqsubswap 12786 | 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 12787 | Closure of square. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | ||
Theorem | sqmul 12788 | Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | ||
Theorem | sqeq0 12789 | A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0)) | ||
Theorem | sqdiv 12790 | Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Mario Carneiro, 9-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | sqdivid 12791 | The square of a nonzero number divided by itself yields the number itself. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) / 𝐴) = 𝐴) | ||
Theorem | sqne0 12792 | A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0)) | ||
Theorem | resqcl 12793 | Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | ||
Theorem | sqgt0 12794 | The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴↑2)) | ||
Theorem | nnsqcl 12795 | The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ∈ ℕ → (𝐴↑2) ∈ ℕ) | ||
Theorem | zsqcl 12796 | Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℤ) | ||
Theorem | qsqcl 12797 | The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | ||
Theorem | sq11 12798 | The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001.) (Proof shortened by Mario Carneiro, 28-May-2016.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵)) | ||
Theorem | lt2sq 12799 | The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) | ||
Theorem | le2sq 12800 | The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2))) |
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