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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | modaddmod 13901 | The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) + 𝐵) mod 𝑀) = ((𝐴 + 𝐵) mod 𝑀)) | ||
Theorem | muladdmodid 13902 | The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.) |
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℝ+ ∧ 𝐴 ∈ (0[,)𝑀)) → (((𝑁 · 𝑀) + 𝐴) mod 𝑀) = 𝐴) | ||
Theorem | mulp1mod1 13903 | The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘2)) → (((𝑁 · 𝐴) + 1) mod 𝑁) = 1) | ||
Theorem | modmuladd 13904* | Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0[,)𝑀) ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 ↔ ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
Theorem | modmuladdim 13905* | Implication of a decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℤ 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
Theorem | modmuladdnn0 13906* | Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵))) | ||
Theorem | negmod 13907 | The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by AV, 5-Jul-2020.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ+) → (-𝐴 mod 𝑁) = ((𝑁 − 𝐴) mod 𝑁)) | ||
Theorem | m1modnnsub1 13908 | Minus one modulo a positive integer is equal to the integer minus one. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝑀 ∈ ℕ → (-1 mod 𝑀) = (𝑀 − 1)) | ||
Theorem | m1modge3gt1 13909 | Minus one modulo an integer greater than two is greater than one. (Contributed by AV, 14-Jul-2021.) |
⊢ (𝑀 ∈ (ℤ≥‘3) → 1 < (-1 mod 𝑀)) | ||
Theorem | addmodid 13910 | The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴) | ||
Theorem | addmodidr 13911 | The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by AV, 19-Mar-2021.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴) | ||
Theorem | modadd2mod 13912 | The sum of a real number modulo a positive real number and another real number equals the sum of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐵 + (𝐴 mod 𝑀)) mod 𝑀) = ((𝐵 + 𝐴) mod 𝑀)) | ||
Theorem | modm1p1mod0 13913 | If a real number modulo a positive real number equals the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals 0. (Contributed by AV, 2-Nov-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = (𝑀 − 1) → ((𝐴 + 1) mod 𝑀) = 0)) | ||
Theorem | modltm1p1mod 13914 | If a real number modulo a positive real number is less than the positive real number decreased by 1, the real number increased by 1 modulo the positive real number equals the real number modulo the positive real number increased by 1. (Contributed by AV, 2-Nov-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ∧ (𝐴 mod 𝑀) < (𝑀 − 1)) → ((𝐴 + 1) mod 𝑀) = ((𝐴 mod 𝑀) + 1)) | ||
Theorem | modmul1 13915 | Multiplication property of the modulo operation. Note that the multiplier 𝐶 must be an integer. (Contributed by NM, 12-Nov-2008.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 · 𝐶) mod 𝐷) = ((𝐵 · 𝐶) mod 𝐷)) | ||
Theorem | modmul12d 13916 | Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ ℤ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸)) | ||
Theorem | modnegd 13917 | Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶)) ⇒ ⊢ (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶)) | ||
Theorem | modadd12d 13918 | Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸)) | ||
Theorem | modsub12d 13919 | Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐶 ∈ ℝ) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸)) & ⊢ (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸)) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐶) mod 𝐸) = ((𝐵 − 𝐷) mod 𝐸)) | ||
Theorem | modsubmod 13920 | The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) − 𝐵) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀)) | ||
Theorem | modsubmodmod 13921 | The difference of a real number modulo a positive real number and another real number modulo this positive real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) − (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 − 𝐵) mod 𝑀)) | ||
Theorem | 2txmodxeq0 13922 | Two times a positive real number modulo the real number is zero. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ (𝑋 ∈ ℝ+ → ((2 · 𝑋) mod 𝑋) = 0) | ||
Theorem | 2submod 13923 | If a real number is between a positive real number and twice the positive real number, the real number modulo the positive real number equals the real number minus the positive real number. (Contributed by Alexander van der Vekens, 13-May-2018.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴 − 𝐵)) | ||
Theorem | modifeq2int 13924 | If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ∧ 𝐴 < (2 · 𝐵)) → (𝐴 mod 𝐵) = if(𝐴 < 𝐵, 𝐴, (𝐴 − 𝐵))) | ||
Theorem | modaddmodup 13925 | The sum of an integer modulo a positive integer and another integer minus the positive integer equals the sum of the two integers modulo the positive integer if the other integer is in the upper part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ ((𝑀 − (𝐴 mod 𝑀))..^𝑀) → ((𝐵 + (𝐴 mod 𝑀)) − 𝑀) = ((𝐵 + 𝐴) mod 𝑀))) | ||
Theorem | modaddmodlo 13926 | The sum of an integer modulo a positive integer and another integer equals the sum of the two integers modulo the positive integer if the other integer is in the lower part of the range between 0 and the positive integer. (Contributed by AV, 30-Oct-2018.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℕ) → (𝐵 ∈ (0..^(𝑀 − (𝐴 mod 𝑀))) → (𝐵 + (𝐴 mod 𝑀)) = ((𝐵 + 𝐴) mod 𝑀))) | ||
Theorem | modmulmod 13927 | The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℤ ∧ 𝑀 ∈ ℝ+) → (((𝐴 mod 𝑀) · 𝐵) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀)) | ||
Theorem | modmulmodr 13928 | The product of an integer and a real number modulo a positive real number equals the product of the integer and the real number modulo the positive real number. (Contributed by Alexander van der Vekens, 9-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → ((𝐴 · (𝐵 mod 𝑀)) mod 𝑀) = ((𝐴 · 𝐵) mod 𝑀)) | ||
Theorem | modaddmulmod 13929 | The sum of a real number and the product of a second real number modulo a positive real number and an integer equals the sum of the real number and the product of the other real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018.) |
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℤ) ∧ 𝑀 ∈ ℝ+) → ((𝐴 + ((𝐵 mod 𝑀) · 𝐶)) mod 𝑀) = ((𝐴 + (𝐵 · 𝐶)) mod 𝑀)) | ||
Theorem | moddi 13930 | Distribute multiplication over a modulo operation. (Contributed by NM, 11-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶))) | ||
Theorem | modsubdir 13931 | Distribute the modulo operation over a subtraction. (Contributed by NM, 30-Dec-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → ((𝐵 mod 𝐶) ≤ (𝐴 mod 𝐶) ↔ ((𝐴 − 𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶)))) | ||
Theorem | modeqmodmin 13932 | A real number equals the difference of the real number and a positive real number modulo the positive real number. (Contributed by AV, 3-Nov-2018.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℝ+) → (𝐴 mod 𝑀) = ((𝐴 − 𝑀) mod 𝑀)) | ||
Theorem | modirr 13933 | A number modulo an irrational multiple of it is nonzero. (Contributed by NM, 11-Nov-2008.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ (𝐴 / 𝐵) ∈ (ℝ ∖ ℚ)) → (𝐴 mod 𝐵) ≠ 0) | ||
Theorem | modfzo0difsn 13934* | For a number within a half-open range of nonnegative integers with one excluded integer there is a positive integer so that the number is equal to the sum of the positive integer and the excluded integer modulo the upper bound of the range. (Contributed by AV, 19-Mar-2021.) |
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑖 ∈ (1..^𝑁)𝐾 = ((𝑖 + 𝐽) mod 𝑁)) | ||
Theorem | modsumfzodifsn 13935 | The sum of a number within a half-open range of positive integers is an element of the corresponding open range of nonnegative integers with one excluded integer modulo the excluded integer. (Contributed by AV, 19-Mar-2021.) |
⊢ ((𝐽 ∈ (0..^𝑁) ∧ 𝐾 ∈ (1..^𝑁)) → ((𝐾 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})) | ||
Theorem | modlteq 13936 | Two nonnegative integers less than the modulus are equal iff they are equal modulo the modulus. (Contributed by AV, 14-Mar-2021.) |
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐼 mod 𝑁) = (𝐽 mod 𝑁) ↔ 𝐼 = 𝐽)) | ||
Theorem | addmodlteq 13937 | Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. A much shorter proof exists if the "divides" relation ∥ can be used, see addmodlteqALT 16295. (Contributed by AV, 20-Mar-2021.) |
⊢ ((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽)) | ||
Theorem | om2uz0i 13938* | The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ0 or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (This series of theorems generalizes an earlier series for ℕ0 contributed by Raph Levien, 10-Apr-2004.) (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ (𝐺‘∅) = 𝐶 | ||
Theorem | om2uzsuci 13939* | The value of 𝐺 (see om2uz0i 13938) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) | ||
Theorem | om2uzuzi 13940* | The value 𝐺 (see om2uz0i 13938) at an ordinal natural number is in the upper integers. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶)) | ||
Theorem | om2uzlti 13941* | Less-than relation for 𝐺 (see om2uz0i 13938). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵))) | ||
Theorem | om2uzlt2i 13942* | The mapping 𝐺 (see om2uz0i 13938) preserves order. (Contributed by NM, 4-May-2005.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) | ||
Theorem | om2uzrani 13943* | Range of 𝐺 (see om2uz0i 13938). (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ ran 𝐺 = (ℤ≥‘𝐶) | ||
Theorem | om2uzf1oi 13944* | 𝐺 (see om2uz0i 13938) is a one-to-one onto mapping. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) | ||
Theorem | om2uzisoi 13945* | 𝐺 (see om2uz0i 13938) is an isomorphism from natural ordinals to upper integers. (Contributed by NM, 9-Oct-2008.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ 𝐺 Isom E , < (ω, (ℤ≥‘𝐶)) | ||
Theorem | om2uzoi 13946* | An alternative definition of 𝐺 in terms of df-oi 9527. (Contributed by Mario Carneiro, 2-Jun-2015.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) ⇒ ⊢ 𝐺 = OrdIso( < , (ℤ≥‘𝐶)) | ||
Theorem | om2uzrdg 13947* | A helper lemma for the value of a recursive definition generator on upper integers (typically either ℕ or ℕ0) with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. Normally 𝐹 is a function on the partition, and 𝐴 is a member of the partition. See also comment in om2uz0i 13938. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) ⇒ ⊢ (𝐵 ∈ ω → (𝑅‘𝐵) = 〈(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))〉) | ||
Theorem | uzrdglem 13948* | A helper lemma for the value of a recursive definition generator on upper integers. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) ⇒ ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) | ||
Theorem | uzrdgfni 13949* | The recursive definition generator on upper integers is a function. See comment in om2uzrdg 13947. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 4-May-2015.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) & ⊢ 𝑆 = ran 𝑅 ⇒ ⊢ 𝑆 Fn (ℤ≥‘𝐶) | ||
Theorem | uzrdg0i 13950* | Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 13947. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) & ⊢ 𝑆 = ran 𝑅 ⇒ ⊢ (𝑆‘𝐶) = 𝐴 | ||
Theorem | uzrdgsuci 13951* | Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 13947. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐶 ∈ ℤ & ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) & ⊢ 𝐴 ∈ V & ⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) & ⊢ 𝑆 = ran 𝑅 ⇒ ⊢ (𝐵 ∈ (ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵))) | ||
Theorem | ltweuz 13952 | < is a well-founded relation on any sequence of upper integers. (Contributed by Andrew Salmon, 13-Nov-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ < We (ℤ≥‘𝐴) | ||
Theorem | ltwenn 13953 | Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.) |
⊢ < We ℕ | ||
Theorem | ltwefz 13954 | Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
⊢ < We (𝑀...𝑁) | ||
Theorem | uzenom 13955 | An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → 𝑍 ≈ ω) | ||
Theorem | uzinf 13956 | An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑀 ∈ ℤ → ¬ 𝑍 ∈ Fin) | ||
Theorem | nnnfi 13957 | The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
⊢ ¬ ℕ ∈ Fin | ||
Theorem | uzrdgxfr 13958* | Transfer the value of the recursive sequence builder from one base to another. (Contributed by Mario Carneiro, 1-Apr-2014.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐴) ↾ ω) & ⊢ 𝐻 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐵) ↾ ω) & ⊢ 𝐴 ∈ ℤ & ⊢ 𝐵 ∈ ℤ ⇒ ⊢ (𝑁 ∈ ω → (𝐺‘𝑁) = ((𝐻‘𝑁) + (𝐴 − 𝐵))) | ||
Theorem | fzennn 13959 | The cardinality of a finite set of sequential integers. (See om2uz0i 13938 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁)) | ||
Theorem | fzen2 13960 | The cardinality of a finite set of sequential integers with arbitrary endpoints. (Contributed by Mario Carneiro, 13-Feb-2014.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀...𝑁) ≈ (◡𝐺‘((𝑁 + 1) − 𝑀))) | ||
Theorem | cardfz 13961 | The cardinality of a finite set of sequential integers. (See om2uz0i 13938 for a description of the hypothesis.) (Contributed by NM, 7-Nov-2008.) (Revised by Mario Carneiro, 15-Sep-2013.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ (𝑁 ∈ ℕ0 → (card‘(1...𝑁)) = (◡𝐺‘𝑁)) | ||
Theorem | hashgf1o 13962 | 𝐺 maps ω one-to-one onto ℕ0. (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) ⇒ ⊢ 𝐺:ω–1-1-onto→ℕ0 | ||
Theorem | fzfi 13963 | A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.) |
⊢ (𝑀...𝑁) ∈ Fin | ||
Theorem | fzfid 13964 | Commonly used special case of fzfi 13963. (Contributed by Mario Carneiro, 25-May-2014.) |
⊢ (𝜑 → (𝑀...𝑁) ∈ Fin) | ||
Theorem | fzofi 13965 | Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
⊢ (𝑀..^𝑁) ∈ Fin | ||
Theorem | fsequb 13966* | The values of a finite real sequence have an upper bound. (Contributed by NM, 19-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
⊢ (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) < 𝑥) | ||
Theorem | fsequb2 13967* | The values of a finite real sequence have an upper bound. (Contributed by NM, 20-Sep-2005.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐹:(𝑀...𝑁)⟶ℝ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝐹 𝑦 ≤ 𝑥) | ||
Theorem | fseqsupcl 13968 | The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ) | ||
Theorem | fseqsupubi 13969 | The values of a finite real sequence are bounded by their supremum. (Contributed by NM, 20-Sep-2005.) |
⊢ ((𝐾 ∈ (𝑀...𝑁) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → (𝐹‘𝐾) ≤ sup(ran 𝐹, ℝ, < )) | ||
Theorem | nn0ennn 13970 | The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.) |
⊢ ℕ0 ≈ ℕ | ||
Theorem | nnenom 13971 | The set of positive integers (as a subset of complex numbers) is equinumerous to omega (the set of finite ordinal numbers). (Contributed by NM, 31-Jul-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
⊢ ℕ ≈ ω | ||
Theorem | nnct 13972 | ℕ is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ ℕ ≼ ω | ||
Theorem | uzindi 13973* | Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ (ℤ≥‘𝐿)) & ⊢ ((𝜑 ∧ 𝑅 ∈ (𝐿...𝑇) ∧ ∀𝑦(𝑆 ∈ (𝐿..^𝑅) → 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆) & ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | axdc4uzlem 13974* | Lemma for axdc4uz 13975. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐴 ∈ V & ⊢ 𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω) & ⊢ 𝐻 = (𝑛 ∈ ω, 𝑥 ∈ 𝐴 ↦ ((𝐺‘𝑛)𝐹𝑥)) ⇒ ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) | ||
Theorem | axdc4uz 13975* | A version of axdc4 10473 that works on an upper set of integers instead of ω. (Contributed by Mario Carneiro, 8-Jan-2014.) |
⊢ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴 ∧ 𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍⟶𝐴 ∧ (𝑔‘𝑀) = 𝐶 ∧ ∀𝑘 ∈ 𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))) | ||
Theorem | ssnn0fi 13976* | A subset of the nonnegative integers is finite if and only if there is a nonnegative integer so that all integers greater than this integer are not contained in the subset. (Contributed by AV, 3-Oct-2019.) |
⊢ (𝑆 ⊆ ℕ0 → (𝑆 ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝑥 ∉ 𝑆))) | ||
Theorem | rabssnn0fi 13977* | A subset of the nonnegative integers defined by a restricted class abstraction is finite if there is a nonnegative integer so that for all integers greater than this integer the condition of the class abstraction is not fulfilled. (Contributed by AV, 3-Oct-2019.) |
⊢ ({𝑥 ∈ ℕ0 ∣ 𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑)) | ||
Theorem | uzsinds 13978* | Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓 → 𝜑)) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜒) | ||
Theorem | nnsinds 13979* | Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 ∈ ℕ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜓 → 𝜑)) ⇒ ⊢ (𝑁 ∈ ℕ → 𝜒) | ||
Theorem | nn0sinds 13980* | Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝑁 → (𝜑 ↔ 𝜒)) & ⊢ (𝑥 ∈ ℕ0 → (∀𝑦 ∈ (0...(𝑥 − 1))𝜓 → 𝜑)) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝜒) | ||
Theorem | fsuppmapnn0fiublem 13981* | Lemma for fsuppmapnn0fiub 13982 and fsuppmapnn0fiubex 13983. (Contributed by AV, 2-Oct-2019.) |
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) & ⊢ 𝑆 = sup(𝑈, ℝ, < ) ⇒ ⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → 𝑆 ∈ ℕ0)) | ||
Theorem | fsuppmapnn0fiub 13982* | If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0 and ending with the supremum of the union of the support of these functions. (Contributed by AV, 2-Oct-2019.) (Proof shortened by JJ, 2-Aug-2021.) |
⊢ 𝑈 = ∪ 𝑓 ∈ 𝑀 (𝑓 supp 𝑍) & ⊢ 𝑆 = sup(𝑈, ℝ, < ) ⇒ ⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → ((∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 ∧ 𝑈 ≠ ∅) → ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆))) | ||
Theorem | fsuppmapnn0fiubex 13983* | If all functions of a finite set of functions over the nonnegative integers are finitely supported, then the support of all these functions is contained in a finite set of sequential integers starting at 0. (Contributed by AV, 2-Oct-2019.) |
⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚))) | ||
Theorem | fsuppmapnn0fiub0 13984* | If all functions of a finite set of functions over the nonnegative integers are finitely supported, then all these functions are zero for all integers greater than a fixed integer. (Contributed by AV, 3-Oct-2019.) |
⊢ ((𝑀 ⊆ (𝑅 ↑m ℕ0) ∧ 𝑀 ∈ Fin ∧ 𝑍 ∈ 𝑉) → (∀𝑓 ∈ 𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑓 ∈ 𝑀 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝑓‘𝑥) = 𝑍))) | ||
Theorem | suppssfz 13985* | Condition for a function over the nonnegative integers to have a support contained in a finite set of sequential integers. (Contributed by AV, 9-Oct-2019.) |
⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0)) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹‘𝑥) = 𝑍)) ⇒ ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆)) | ||
Theorem | fsuppmapnn0ub 13986* | If a function over the nonnegative integers is finitely supported, then there is an upper bound for the arguments resulting in nonzero values. (Contributed by AV, 6-Oct-2019.) |
⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹‘𝑥) = 𝑍))) | ||
Theorem | fsuppmapnn0fz 13987* | If a function over the nonnegative integers is finitely supported, then there is an upper bound for a finite set of sequential integers containing the support of the function. (Contributed by AV, 30-Sep-2019.) (Proof shortened by AV, 6-Oct-2019.) |
⊢ ((𝐹 ∈ (𝑅 ↑m ℕ0) ∧ 𝑍 ∈ 𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 (𝐹 supp 𝑍) ⊆ (0...𝑚))) | ||
Theorem | mptnn0fsupp 13988* | A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.) |
⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) ⇒ ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) | ||
Theorem | mptnn0fsuppd 13989* | A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.) |
⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) & ⊢ (𝑘 = 𝑥 → 𝐶 = 𝐷) & ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → 𝐷 = 0 )) ⇒ ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) | ||
Theorem | mptnn0fsuppr 13990* | A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 23-Dec-2019.) |
⊢ (𝜑 → 0 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) finSupp 0 ) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )) | ||
Theorem | f13idfv 13991 | A one-to-one function with the domain { 0, 1 ,2 } in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ 𝐴 = (0...2) ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ((𝐹‘0) ≠ (𝐹‘1) ∧ (𝐹‘0) ≠ (𝐹‘2) ∧ (𝐹‘1) ≠ (𝐹‘2)))) | ||
Syntax | cseq 13992 | Extend class notation with recursive sequence builder. |
class seq𝑀( + , 𝐹) | ||
Definition | df-seq 13993* |
Define a general-purpose operation that builds a recursive sequence
(i.e., a function on an upper integer set such as ℕ or ℕ0)
whose value at an index is a function of its previous value and the
value of an input sequence at that index. This definition is
complicated, but fortunately it is not intended to be used directly.
Instead, the only purpose of this definition is to provide us with an
object that has the properties expressed by seq1 14005
and seqp1 14007.
Typically, those are the main theorems that would be used in practice.
The first operand in the parentheses is the operation that is applied to the previous value and the value of the input sequence (second operand). The operand to the left of the parenthesis is the integer to start from. For example, for the operation +, an input sequence 𝐹 with values 1, 1/2, 1/4, 1/8,... would be transformed into the output sequence seq1( + , 𝐹) with values 1, 3/2, 7/4, 15/8,.., so that (seq1( + , 𝐹)‘1) = 1, (seq1( + , 𝐹)‘2) = 3/2, etc. In other words, seq𝑀( + , 𝐹) transforms a sequence 𝐹 into an infinite series. seq𝑀( + , 𝐹) ⇝ 2 means "the sum of F(n) from n = M to infinity is 2". Since limits are unique (climuni 15522), by climdm 15524 the "sum of F(n) from n = 1 to infinity" can be expressed as ( ⇝ ‘seq1( + , 𝐹)) (provided the sequence converges) and evaluates to 2 in this example. Internally, the rec function generates as its values a set of ordered pairs starting at 〈𝑀, (𝐹‘𝑀)〉, with the first member of each pair incremented by one in each successive value. So, the range of rec is exactly the sequence we want, and we just extract the range (restricted to omega) and throw away the domain. This definition has its roots in a series of theorems from om2uz0i 13938 through om2uzf1oi 13944, originally proved by Raph Levien for use with df-exp 14053 and later generalized for arbitrary recursive sequences. Definition df-sum 15659 extracts the summation values from partial (finite) and complete (infinite) series. (Contributed by NM, 18-Apr-2005.) (Revised by Mario Carneiro, 4-Sep-2013.) |
⊢ seq𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) | ||
Theorem | seqex 13994 | Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
⊢ seq𝑀( + , 𝐹) ∈ V | ||
Theorem | seqeq1 13995 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
⊢ (𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹)) | ||
Theorem | seqeq2 13996 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) | ||
Theorem | seqeq3 13997 | Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
⊢ (𝐹 = 𝐺 → seq𝑀( + , 𝐹) = seq𝑀( + , 𝐺)) | ||
Theorem | seqeq1d 13998 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → seq𝐴( + , 𝐹) = seq𝐵( + , 𝐹)) | ||
Theorem | seqeq2d 13999 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → seq𝑀(𝐴, 𝐹) = seq𝑀(𝐵, 𝐹)) | ||
Theorem | seqeq3d 14000 | Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → seq𝑀( + , 𝐴) = seq𝑀( + , 𝐵)) |
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