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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | expnnval 14101 | 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 14102 | Value of a complex number raised to the zeroth power. Under our definition, 0↑0 = 1 (0exp0e1 14103), following standard convention, for instance 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 14103 | The zeroth power of zero equals one. See comment of exp0 14102. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0↑0) = 1 | ||
Theorem | exp1 14104 | 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 14105 | Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of [Gleason] p. 134. When 𝐴 is nonzero, this holds for all integers 𝑁, see expneg 14106. (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 14106 | Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expneg2 14107 | Value of a complex number raised to a nonpositive integer power. When 𝐴 = 0 and 𝑁 is nonzero, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) | ||
Theorem | expn1 14108 | A complex number raised to the negative one power is its reciprocal. When 𝐴 = 0, both sides have the "value" (1 / 0); relying on that should be avoid in applications. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑-1) = (1 / 𝐴)) | ||
Theorem | expcllem 14109* | 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 14110* | 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 14111 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) | ||
Theorem | nn0expcl 14112 | Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ0) | ||
Theorem | zexpcl 14113 | Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℤ) | ||
Theorem | qexpcl 14114 | Closure of exponentiation of rationals. For integer exponents, see qexpclz 14118. (Contributed by NM, 16-Dec-2005.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | reexpcl 14115 | Closure of exponentiation of reals. For integer exponents, see reexpclz 14119. (Contributed by NM, 14-Dec-2005.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expcl 14116 | Closure law for nonnegative integer exponentiation. For integer exponents, see expclz 14121. (Contributed by NM, 26-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | rpexpcl 14117 | Closure law for integer 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 | qexpclz 14118 | Closure of integer exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.) |
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℚ) | ||
Theorem | reexpclz 14119 | Closure of integer 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 | expclzlem 14120 | Lemma for expclz 14121. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ (ℂ ∖ {0})) | ||
Theorem | expclz 14121 | Closure law for integer exponentiation of complex numnbers. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | m1expcl2 14122 | Closure of integer exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ {-1, 1}) | ||
Theorem | m1expcl 14123 | Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑁 ∈ ℤ → (-1↑𝑁) ∈ ℤ) | ||
Theorem | zexpcld 14124 | Closure of exponentiation of integers, deduction form. (Contributed by SN, 15-Sep-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℤ) | ||
Theorem | nn0expcli 14125 | Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.) |
⊢ 𝐴 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 ⇒ ⊢ (𝐴↑𝑁) ∈ ℕ0 | ||
Theorem | nn0sqcl 14126 | 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 14127 | Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = ((𝐴↑(𝑁 − 1)) · 𝐴)) | ||
Theorem | 1exp 14128 | Value of 1 raised to an integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) | ||
Theorem | expeq0 14129 | A positive integer power is zero if and only if its base is zero. (Contributed by NM, 23-Feb-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) = 0 ↔ 𝐴 = 0)) | ||
Theorem | expne0 14130 | A positive integer power is nonzero if and only if its base is nonzero. (Contributed by NM, 6-May-2005.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((𝐴↑𝑁) ≠ 0 ↔ 𝐴 ≠ 0)) | ||
Theorem | expne0i 14131 | An integer power is nonzero if its base 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 14132 | A positive real raised to an integer power 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 14133 | Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 4-Jun-2014.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | 0exp 14134 | Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.) |
⊢ (𝑁 ∈ ℕ → (0↑𝑁) = 0) | ||
Theorem | expge0 14135 | A nonnegative real raised to a nonnegative integer 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 14136 | A real greater than or equal to 1 raised to a nonnegative integer 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 14137 | A real greater than 1 raised to a positive integer 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 14138 | Nonnegative 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 14139 | 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 14140 | 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 14141 | 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 14142 | Lemma for expaddz 14143. (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 14143 | 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 14144 | Product of exponents law for nonnegative 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 14145 | 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 14146 | Exponentiation of negative one to an even power. (Contributed by Scott Fenton, 17-Jan-2018.) |
⊢ (𝑁 ∈ ℤ → (-1↑(2 · 𝑁)) = 1) | ||
Theorem | expsub 14147 | Exponent subtraction law for 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 14148 | 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 14149 | Value of a nonzero 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 14150 | 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 | sqval 14151 | Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | ||
Theorem | sqneg 14152 | The square of the negative of a number. (Contributed by NM, 15-Jan-2006.) |
⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2)) | ||
Theorem | sqsubswap 14153 | 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 14154 | Closure of square. (Contributed by NM, 10-Aug-1999.) |
⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | ||
Theorem | sqmul 14155 | Distribution of squaring over multiplication. (Contributed by NM, 21-Mar-2008.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | ||
Theorem | sqeq0 14156 | A complex number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) = 0 ↔ 𝐴 = 0)) | ||
Theorem | sqdiv 14157 | Distribution of squaring 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 14158 | The square of a nonzero complex number divided by itself equals that number. (Contributed by AV, 19-Jul-2021.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴↑2) / 𝐴) = 𝐴) | ||
Theorem | sqne0 14159 | A complex number is nonzero if and only if its square is nonzero. (Contributed by NM, 11-Mar-2006.) |
⊢ (𝐴 ∈ ℂ → ((𝐴↑2) ≠ 0 ↔ 𝐴 ≠ 0)) | ||
Theorem | resqcl 14160 | Closure of squaring in reals. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝐴 ∈ ℝ → (𝐴↑2) ∈ ℝ) | ||
Theorem | resqcld 14161 | Closure of squaring in reals, deduction form. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℝ) | ||
Theorem | sqgt0 14162 | The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴↑2)) | ||
Theorem | sqn0rp 14163 | The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (𝐴↑2) ∈ ℝ+) | ||
Theorem | nnsqcl 14164 | The positive 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 14165 | 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 14166 | The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.) |
⊢ (𝐴 ∈ ℚ → (𝐴↑2) ∈ ℚ) | ||
Theorem | sq11 14167 | 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 | nn0sq11 14168 | The square function is one-to-one for nonnegative integers. (Contributed by AV, 25-Jun-2023.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵)) | ||
Theorem | lt2sq 14169 | The square function is increasing on nonnegative reals. (Contributed by NM, 24-Feb-2006.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 < 𝐵 ↔ (𝐴↑2) < (𝐵↑2))) | ||
Theorem | le2sq 14170 | The square function is nondecreasing on nonnegative reals. (Contributed by NM, 18-Oct-1999.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (𝐴 ≤ 𝐵 ↔ (𝐴↑2) ≤ (𝐵↑2))) | ||
Theorem | le2sq2 14171 | The square function is nondecreasing on the nonnegative reals. (Contributed by NM, 21-Mar-2008.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵)) → (𝐴↑2) ≤ (𝐵↑2)) | ||
Theorem | sqge0 14172 | The square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.) |
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴↑2)) | ||
Theorem | sqge0d 14173 | The square of a real is nonnegative, deduction form. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 0 ≤ (𝐴↑2)) | ||
Theorem | zsqcl2 14174 | The square of an integer is a nonnegative integer. (Contributed by Mario Carneiro, 18-Apr-2014.) (Revised by Mario Carneiro, 14-Jul-2014.) |
⊢ (𝐴 ∈ ℤ → (𝐴↑2) ∈ ℕ0) | ||
Theorem | 0expd 14175 | Value of zero raised to a positive integer power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (0↑𝑁) = 0) | ||
Theorem | exp0d 14176 | Value of a complex number raised to the zeroth power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑0) = 1) | ||
Theorem | exp1d 14177 | 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 14178 | If a positive integer power is zero, then its base is zero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (𝐴↑𝑁) = 0) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | sqvald 14179 | Value of square. Inference version. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑2) = (𝐴 · 𝐴)) | ||
Theorem | sqcld 14180 | Closure of square. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) | ||
Theorem | sqeq0d 14181 | 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 14182 | 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 14183 | 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 14184 | 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 14185 | 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 | sqrecd 14186 | Square of reciprocal is reciprocal of square. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑2) = (1 / (𝐴↑2))) | ||
Theorem | expclzd 14187 | Closure law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) | ||
Theorem | expne0d 14188 | A nonnegative integer power is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) | ||
Theorem | expnegd 14189 | Value of a nonzero complex number raised to the negative of an integer power. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑-𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | exprecd 14190 | An integer power of a reciprocal is the reciprocal of the integer power with same exponent. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁))) | ||
Theorem | expp1zd 14191 | Value of a nonzero complex number raised to an integer power plus one. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 + 1)) = ((𝐴↑𝑁) · 𝐴)) | ||
Theorem | expm1d 14192 | Value of a nonzero complex number raised to an integer power minus one. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴)) | ||
Theorem | expsubd 14193 | Exponent subtraction law for integer exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐴↑(𝑀 − 𝑁)) = ((𝐴↑𝑀) / (𝐴↑𝑁))) | ||
Theorem | sqmuld 14194 | Distribution of squaring over multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | ||
Theorem | sqdivd 14195 | Distribution of squaring over division. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | expdivd 14196 | Nonnegative integer exponentiation of a quotient. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴 / 𝐵)↑𝑁) = ((𝐴↑𝑁) / (𝐵↑𝑁))) | ||
Theorem | mulexpd 14197 | Nonnegative 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 | znsqcld 14198 | The square of a nonzero integer is a positive integer. (Contributed by Thierry Arnoux, 2-Feb-2020.) |
⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ≠ 0) ⇒ ⊢ (𝜑 → (𝑁↑2) ∈ ℕ) | ||
Theorem | reexpcld 14199 | Closure of exponentiation of reals. (Contributed by Mario Carneiro, 28-May-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) | ||
Theorem | expge0d 14200 | 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)) |
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