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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | binom2sub 10601 | Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2))) | ||
Theorem | binom2sub1 10602 | Special case of binom2sub 10601 where 𝐵 = 1. (Contributed by AV, 2-Aug-2021.) |
⊢ (𝐴 ∈ ℂ → ((𝐴 − 1)↑2) = (((𝐴↑2) − (2 · 𝐴)) + 1)) | ||
Theorem | binom2subi 10603 | Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((𝐴 − 𝐵)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝐵))) + (𝐵↑2)) | ||
Theorem | mulbinom2 10604 | The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (((𝐶 · 𝐴) + 𝐵)↑2) = ((((𝐶 · 𝐴)↑2) + ((2 · 𝐶) · (𝐴 · 𝐵))) + (𝐵↑2))) | ||
Theorem | binom3 10605 | The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑3) = (((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3)))) | ||
Theorem | zesq 10606 | An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.) |
⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ↔ ((𝑁↑2) / 2) ∈ ℤ)) | ||
Theorem | nnesq 10607 | A positive integer is even iff its square is even. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) |
⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ↔ ((𝑁↑2) / 2) ∈ ℕ)) | ||
Theorem | bernneq 10608 | Bernoulli's inequality, due to Johan Bernoulli (1667-1748). (Contributed by NM, 21-Feb-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ 𝐴) → (1 + (𝐴 · 𝑁)) ≤ ((1 + 𝐴)↑𝑁)) | ||
Theorem | bernneq2 10609 | Variation of Bernoulli's inequality bernneq 10608. (Contributed by NM, 18-Oct-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁)) | ||
Theorem | bernneq3 10610 | A corollary of bernneq 10608. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ 𝑁 ∈ ℕ0) → 𝑁 < (𝑃↑𝑁)) | ||
Theorem | expnbnd 10611* | Exponentiation with a base greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ 𝐴 < (𝐵↑𝑘)) | ||
Theorem | expnlbnd 10612* | The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑘 ∈ ℕ (1 / (𝐵↑𝑘)) < 𝐴) | ||
Theorem | expnlbnd2 10613* | The reciprocal of exponentiation with a base greater than 1 has no positive lower bound. (Contributed by NM, 18-Jul-2008.) (Proof shortened by Mario Carneiro, 5-Jun-2014.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 1 < 𝐵) → ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)(1 / (𝐵↑𝑘)) < 𝐴) | ||
Theorem | modqexp 10614 | Exponentiation property of the modulo operation, see theorem 5.2(c) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by Jim Kingdon, 7-Sep-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℚ) & ⊢ (𝜑 → 0 < 𝐷) & ⊢ (𝜑 → (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) ⇒ ⊢ (𝜑 → ((𝐴↑𝐶) mod 𝐷) = ((𝐵↑𝐶) mod 𝐷)) | ||
Theorem | exp0d 10615 | Value of a complex number raised to the 0th power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑0) = 1) | ||
Theorem | exp1d 10616 | Value of a complex number raised to the first power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑1) = 𝐴) | ||
Theorem | expeq0d 10617 | Positive integer exponentiation is 0 iff its base is 0. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐴↑𝑁) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | sqvald 10618 | Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) | ||
Theorem | sqcld 10619 | Closure of square. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) | ||
Theorem | sqeq0d 10620 | A number is zero iff its square is zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (𝐴↑2) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | expcld 10621 | Closure law for nonnegative integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | expp1d 10622 | 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 Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expaddd 10623 | Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑(𝑀 + 𝑁)) = ((𝐴↑𝑀) · (𝐴↑𝑁))) | ||
Theorem | expmuld 10624 | Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑(𝑀 · 𝑁)) = ((𝐴↑𝑀)↑𝑁)) | ||
Theorem | sqrecapd 10625 | Square of reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2))) | ||
Theorem | expclzapd 10626 | Closure law for integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | expap0d 10627 | Nonnegative integer exponentiation is nonzero if its base is nonzero. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) # 0) | ||
Theorem | expnegapd 10628 | Value of a complex number raised to a negative power. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | exprecapd 10629 | Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expp1zapd 10630 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expm1apd 10631 | Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴)) | ||
Theorem | expsubapd 10632 | Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 12-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁))) | ||
Theorem | sqmuld 10633 | Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | ||
Theorem | sqdivapd 10634 | Distribution of square over division. (Contributed by Jim Kingdon, 13-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | expdivapd 10635 | Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 13-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁))) | ||
Theorem | mulexpd 10636 | Positive integer exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑𝑁) = ((𝐴↑𝑁) · (𝐵↑𝑁))) | ||
Theorem | 0expd 10637 | Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (0↑𝑁) = 0) | ||
Theorem | reexpcld 10638 | Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expge0d 10639 | A nonnegative real raised to a nonnegative integer is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 0 ≤ 𝐴) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴↑𝑁)) | ||
Theorem | expge1d 10640 | A real greater than or equal to 1 raised to a nonnegative integer is greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 1 ≤ 𝐴) ⇒ ⊢ (𝜑 → 1 ≤ (𝐴↑𝑁)) | ||
Theorem | sqoddm1div8 10641 | A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 = ((2 · 𝑁) + 1)) → (((𝑀↑2) − 1) / 8) = ((𝑁 · (𝑁 + 1)) / 2)) | ||
Theorem | nnsqcld 10642 | The naturals are closed under squaring. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) | ||
Theorem | nnexpcld 10643 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcld 10644 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | rpexpcld 10645 | Closure law for exponentiation of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ+) | ||
Theorem | reexpclzapd 10646 | Closure of exponentiation of reals. (Contributed by Jim Kingdon, 13-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | resqcld 10647 | Closure of square in reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℝ) | ||
Theorem | sqge0d 10648 | A square of a real is nonnegative. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴↑2)) | ||
Theorem | sqgt0apd 10649 | The square of a real apart from zero is positive. (Contributed by Jim Kingdon, 13-Jun-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 # 0) ⇒ ⊢ (𝜑 → 0 < (𝐴↑2)) | ||
Theorem | leexp2ad 10650 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 1 ≤ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) | ||
Theorem | leexp2rd 10651 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 1) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ≤ (𝐴↑𝑀)) | ||
Theorem | lt2sqd 10652 | The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) | ||
Theorem | le2sqd 10653 | The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2))) | ||
Theorem | sq11d 10654 | The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 0 ≤ 𝐵) & ⊢ (𝜑 → (𝐴↑2) = (𝐵↑2)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
Theorem | sq11ap 10655 | Analogue to sq11 10560 but for apartness. (Contributed by Jim Kingdon, 12-Aug-2021.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) # (𝐵↑2) ↔ 𝐴 # 𝐵)) | ||
Theorem | nn0ltexp2 10656 | Special case of ltexp2 13911 which we use here because we haven't yet defined df-rpcxp 13831 which is used in the current proof of ltexp2 13911. (Contributed by Jim Kingdon, 7-Oct-2024.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) | ||
Theorem | nn0leexp2 10657 | Ordering law for exponentiation. (Contributed by Jim Kingdon, 9-Oct-2024.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁))) | ||
Theorem | sq10 10658 | The square of 10 is 100. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = ;;100 | ||
Theorem | sq10e99m1 10659 | The square of 10 is 99 plus 1. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ (;10↑2) = (;99 + 1) | ||
Theorem | 3dec 10660 | A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 ⇒ ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) | ||
Theorem | expcanlem 10661 | Lemma for expcan 10662. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) ⇒ ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁)) | ||
Theorem | expcan 10662 | 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 | expcand 10663 | Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → (𝐴↑𝑀) = (𝐴↑𝑁)) ⇒ ⊢ (𝜑 → 𝑀 = 𝑁) | ||
Theorem | apexp1 10664 | Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) # (𝐵↑𝑁) → 𝐴 # 𝐵)) | ||
Theorem | nn0le2msqd 10665 | The square function on nonnegative integers is monotonic. (Contributed by Jim Kingdon, 31-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ (𝐴 · 𝐴) ≤ (𝐵 · 𝐵))) | ||
Theorem | nn0opthlem1d 10666 | A rather pretty lemma for nn0opth2 10670. (Contributed by Jim Kingdon, 31-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴 < 𝐶 ↔ ((𝐴 · 𝐴) + (2 · 𝐴)) < (𝐶 · 𝐶))) | ||
Theorem | nn0opthlem2d 10667 | Lemma for nn0opth2 10670. (Contributed by Jim Kingdon, 31-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) < 𝐶 → ((𝐶 · 𝐶) + 𝐷) ≠ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵))) | ||
Theorem | nn0opthd 10668 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers 𝐴 and 𝐵 by (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3598 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | nn0opth2d 10669 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 10668. (Contributed by Jim Kingdon, 31-Oct-2021.) |
⊢ (𝜑 → 𝐴 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Theorem | nn0opth2 10670 | An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthd 10668. (Contributed by NM, 22-Jul-2004.) |
⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Syntax | cfa 10671 | Extend class notation to include the factorial of nonnegative integers. |
class ! | ||
Definition | df-fac 10672 | Define the factorial function on nonnegative integers. For example, (!‘5) = 120 because 1 · 2 · 3 · 4 · 5 = 120 (ex-fac 14020). In the literature, the factorial function is written as a postscript exclamation point. (Contributed by NM, 2-Dec-2004.) |
⊢ ! = ({〈0, 1〉} ∪ seq1( · , I )) | ||
Theorem | facnn 10673 | Value of the factorial function for positive integers. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = (seq1( · , I )‘𝑁)) | ||
Theorem | fac0 10674 | The factorial of 0. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (!‘0) = 1 | ||
Theorem | fac1 10675 | The factorial of 1. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (!‘1) = 1 | ||
Theorem | facp1 10676 | The factorial of a successor. (Contributed by NM, 2-Dec-2004.) (Revised by Mario Carneiro, 13-Jul-2013.) |
⊢ (𝑁 ∈ ℕ0 → (!‘(𝑁 + 1)) = ((!‘𝑁) · (𝑁 + 1))) | ||
Theorem | fac2 10677 | The factorial of 2. (Contributed by NM, 17-Mar-2005.) |
⊢ (!‘2) = 2 | ||
Theorem | fac3 10678 | The factorial of 3. (Contributed by NM, 17-Mar-2005.) |
⊢ (!‘3) = 6 | ||
Theorem | fac4 10679 | The factorial of 4. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (!‘4) = ;24 | ||
Theorem | facnn2 10680 | Value of the factorial function expressed recursively. (Contributed by NM, 2-Dec-2004.) |
⊢ (𝑁 ∈ ℕ → (!‘𝑁) = ((!‘(𝑁 − 1)) · 𝑁)) | ||
Theorem | faccl 10681 | Closure of the factorial function. (Contributed by NM, 2-Dec-2004.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | ||
Theorem | faccld 10682 | Closure of the factorial function, deduction version of faccl 10681. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (!‘𝑁) ∈ ℕ) | ||
Theorem | facne0 10683 | The factorial function is nonzero. (Contributed by NM, 26-Apr-2005.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≠ 0) | ||
Theorem | facdiv 10684 | A positive integer divides the factorial of an equal or larger number. (Contributed by NM, 2-May-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → ((!‘𝑀) / 𝑁) ∈ ℕ) | ||
Theorem | facndiv 10685 | No positive integer (greater than one) divides the factorial plus one of an equal or larger number. (Contributed by NM, 3-May-2005.) |
⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) ∧ (1 < 𝑁 ∧ 𝑁 ≤ 𝑀)) → ¬ (((!‘𝑀) + 1) / 𝑁) ∈ ℤ) | ||
Theorem | facwordi 10686 | Ordering property of factorial. (Contributed by NM, 9-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → (!‘𝑀) ≤ (!‘𝑁)) | ||
Theorem | faclbnd 10687 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑(𝑁 + 1)) ≤ ((𝑀↑𝑀) · (!‘𝑁))) | ||
Theorem | faclbnd2 10688 | A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁)) | ||
Theorem | faclbnd3 10689 | A lower bound for the factorial function. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀↑𝑁) ≤ ((𝑀↑𝑀) · (!‘𝑁))) | ||
Theorem | faclbnd6 10690 | Geometric lower bound for the factorial function, where N is usually held constant. (Contributed by Paul Chapman, 28-Dec-2007.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → ((!‘𝑁) · ((𝑁 + 1)↑𝑀)) ≤ (!‘(𝑁 + 𝑀))) | ||
Theorem | facubnd 10691 | An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.) |
⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ≤ (𝑁↑𝑁)) | ||
Theorem | facavg 10692 | The product of two factorials is greater than or equal to the factorial of (the floor of) their average. (Contributed by NM, 9-Dec-2005.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (!‘(⌊‘((𝑀 + 𝑁) / 2))) ≤ ((!‘𝑀) · (!‘𝑁))) | ||
Syntax | cbc 10693 | Extend class notation to include the binomial coefficient operation (combinatorial choose operation). |
class C | ||
Definition | df-bc 10694* |
Define the binomial coefficient operation. For example,
(5C3) = 10 (ex-bc 14021).
In the literature, this function is often written as a column vector of the two arguments, or with the arguments as subscripts before and after the letter "C". (𝑁C𝐾) is read "𝑁 choose 𝐾." Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝑘 ≤ 𝑛 does not hold. (Contributed by NM, 10-Jul-2005.) |
⊢ C = (𝑛 ∈ ℕ0, 𝑘 ∈ ℤ ↦ if(𝑘 ∈ (0...𝑛), ((!‘𝑛) / ((!‘(𝑛 − 𝑘)) · (!‘𝑘))), 0)) | ||
Theorem | bcval 10695 | Value of the binomial coefficient, 𝑁 choose 𝐾. Definition of binomial coefficient in [Gleason] p. 295. As suggested by Gleason, we define it to be 0 when 0 ≤ 𝐾 ≤ 𝑁 does not hold. See bcval2 10696 for the value in the standard domain. (Contributed by NM, 10-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = if(𝐾 ∈ (0...𝑁), ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾))), 0)) | ||
Theorem | bcval2 10696 | Value of the binomial coefficient, 𝑁 choose 𝐾, in its standard domain. (Contributed by NM, 9-Jun-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) = ((!‘𝑁) / ((!‘(𝑁 − 𝐾)) · (!‘𝐾)))) | ||
Theorem | bcval3 10697 | Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 8-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ ¬ 𝐾 ∈ (0...𝑁)) → (𝑁C𝐾) = 0) | ||
Theorem | bcval4 10698 | Value of the binomial coefficient, 𝑁 choose 𝐾, outside of its standard domain. Remark in [Gleason] p. 295. (Contributed by NM, 14-Jul-2005.) (Revised by Mario Carneiro, 7-Nov-2013.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ ∧ (𝐾 < 0 ∨ 𝑁 < 𝐾)) → (𝑁C𝐾) = 0) | ||
Theorem | bcrpcl 10699 | Closure of the binomial coefficient in the positive reals. (This is mostly a lemma before we have bccl2 10714.) (Contributed by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝐾 ∈ (0...𝑁) → (𝑁C𝐾) ∈ ℝ+) | ||
Theorem | bccmpl 10700 | "Complementing" its second argument doesn't change a binary coefficient. (Contributed by NM, 21-Jun-2005.) (Revised by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → (𝑁C𝐾) = (𝑁C(𝑁 − 𝐾))) |
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