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Theorem List for Metamath Proof Explorer - 12701-12800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmodcyc2 12701 The modulo operation is periodic. (Contributed by NM, 12-Nov-2008.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+𝑁 ∈ ℤ) → ((𝐴 − (𝐵 · 𝑁)) mod 𝐵) = (𝐴 mod 𝐵))

Theoremmodadd1 12702 Addition property of the modulo operation. (Contributed by NM, 12-Nov-2008.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ+) ∧ (𝐴 mod 𝐷) = (𝐵 mod 𝐷)) → ((𝐴 + 𝐶) mod 𝐷) = ((𝐵 + 𝐶) mod 𝐷))

Theoremmodaddabs 12703 Absorption law for modulo. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ+) → (((𝐴 mod 𝐶) + (𝐵 mod 𝐶)) mod 𝐶) = ((𝐴 + 𝐵) mod 𝐶))

Theoremmodaddmod 12704 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))

Theoremmuladdmodid 12705 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)

Theoremmulp1mod1 12706 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)

Theoremmodmuladd 12707* 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)))

Theoremmodmuladdim 12708* 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)))

Theoremmodmuladdnn0 12709* Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)
((𝐴 ∈ ℕ0𝑀 ∈ ℝ+) → ((𝐴 mod 𝑀) = 𝐵 → ∃𝑘 ∈ ℕ0 𝐴 = ((𝑘 · 𝑀) + 𝐵)))

Theoremnegmod 12710 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) = ((𝑁𝐴) mod 𝑁))

Theoremm1modnnsub1 12711 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))

Theoremm1modge3gt1 12712 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))

Theoremaddmodid 12713 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 ∈ ℕ0𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝑀 + 𝐴) mod 𝑀) = 𝐴)

Theoremaddmodidr 12714 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 ∈ ℕ0𝑀 ∈ ℕ ∧ 𝐴 < 𝑀) → ((𝐴 + 𝑀) mod 𝑀) = 𝐴)

Theoremmodadd2mod 12715 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))

Theoremmodm1p1mod0 12716 If an 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))

Theoremmodltm1p1mod 12717 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))

Theoremmodmul1 12718 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))

Theoremmodmul12d 12719 Multiplication property of the modulo operation, see theorem 5.2(b) in [ApostolNT] p. 107. (Contributed by Mario Carneiro, 5-Feb-2015.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℤ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 · 𝐶) mod 𝐸) = ((𝐵 · 𝐷) mod 𝐸))

Theoremmodnegd 12720 Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐶) = (𝐵 mod 𝐶))       (𝜑 → (-𝐴 mod 𝐶) = (-𝐵 mod 𝐶))

Theoremmodadd12d 12721 Additive property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴 + 𝐶) mod 𝐸) = ((𝐵 + 𝐷) mod 𝐸))

Theoremmodsub12d 12722 Subtraction property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (𝐴 mod 𝐸) = (𝐵 mod 𝐸))    &   (𝜑 → (𝐶 mod 𝐸) = (𝐷 mod 𝐸))       (𝜑 → ((𝐴𝐶) mod 𝐸) = ((𝐵𝐷) mod 𝐸))

Theoremmodsubmod 12723 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) = ((𝐴𝐵) mod 𝑀))

Theoremmodsubmodmod 12724 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) = ((𝐴𝐵) mod 𝑀))

Theorem2txmodxeq0 12725 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)

Theorem2submod 12726 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+) ∧ (𝐵𝐴𝐴 < (2 · 𝐵))) → (𝐴 mod 𝐵) = (𝐴𝐵))

Theoremmodifeq2int 12727 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 ∈ ℕ0𝐵 ∈ ℕ ∧ 𝐴 < (2 · 𝐵)) → (𝐴 mod 𝐵) = if(𝐴 < 𝐵, 𝐴, (𝐴𝐵)))

Theoremmodaddmodup 12728 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)))

Theoremmodaddmodlo 12729 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)))

Theoremmodmulmod 12730 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))

Theoremmodmulmodr 12731 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))

Theoremmodaddmulmod 12732 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))

Theoremmoddi 12733 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)))

Theoremmodsubdir 12734 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) ↔ ((𝐴𝐵) mod 𝐶) = ((𝐴 mod 𝐶) − (𝐵 mod 𝐶))))

Theoremmodeqmodmin 12735 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) = ((𝐴𝑀) mod 𝑀))

Theoremmodirr 12736 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)

Theoremmodfzo0difsn 12737* 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))

Theoremmodsumfzodifsn 12738 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}))

Theoremmodlteq 12739 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))

Theoremaddmodlteq 12740 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 15041. (Contributed by AV, 20-Mar-2021.)
((𝐼 ∈ (0..^𝑁) ∧ 𝐽 ∈ (0..^𝑁) ∧ 𝑆 ∈ ℤ) → (((𝐼 + 𝑆) mod 𝑁) = ((𝐽 + 𝑆) mod 𝑁) ↔ 𝐼 = 𝐽))

Theoremom2uz0i 12741* 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 0 or 1 for the upper integers ), 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 0 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

Theoremom2uzsuci 12742* The value of 𝐺 (see om2uz0i 12741) 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) = ((𝐺𝐴) + 1))

Theoremom2uzuzi 12743* The value 𝐺 (see om2uz0i 12741) 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 → (𝐺𝐴) ∈ (ℤ𝐶))

Theoremom2uzlti 12744* Less-than relation for 𝐺 (see om2uz0i 12741). (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) → (𝐴𝐵 → (𝐺𝐴) < (𝐺𝐵)))

Theoremom2uzlt2i 12745* The mapping 𝐺 (see om2uz0i 12741) 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) → (𝐴𝐵 ↔ (𝐺𝐴) < (𝐺𝐵)))

Theoremom2uzrani 12746* Range of 𝐺 (see om2uz0i 12741). (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 = (ℤ𝐶)

Theoremom2uzf1oi 12747* 𝐺 (see om2uz0i 12741) 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→(ℤ𝐶)

Theoremom2uzisoi 12748* 𝐺 (see om2uz0i 12741) 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, (ℤ𝐶))

Theoremom2uzoi 12749* An alternative definition of 𝐺 in terms of df-oi 8412. (Contributed by Mario Carneiro, 2-Jun-2015.)
𝐶 ∈ ℤ    &   𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω)       𝐺 = OrdIso( < , (ℤ𝐶))

Theoremom2uzrdg 12750* A helper lemma for the value of a recursive definition generator on upper integers (typically either or 0) 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 12741. (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 → (𝑅𝐵) = ⟨(𝐺𝐵), (2nd ‘(𝑅𝐵))⟩)

Theoremuzrdglem 12751* 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 ∈ (ℤ𝐶) → ⟨𝐵, (2nd ‘(𝑅‘(𝐺𝐵)))⟩ ∈ ran 𝑅)

Theoremuzrdgfni 12752* The recursive definition generator on upper integers is a function. See comment in om2uzrdg 12750. (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 (ℤ𝐶)

Theoremuzrdg0i 12753* Initial value of a recursive definition generator on upper integers. See comment in om2uzrdg 12750. (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       (𝑆𝐶) = 𝐴

Theoremuzrdgsuci 12754* Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12750. (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 ∈ (ℤ𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆𝐵)))

Theoremltweuz 12755 < 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 (ℤ𝐴)

Theoremltwenn 12756 Less than well-orders the naturals. (Contributed by Scott Fenton, 6-Aug-2013.)
< We ℕ

Theoremltwefz 12757 Less than well-orders a set of finite integers. (Contributed by Scott Fenton, 8-Aug-2013.)
< We (𝑀...𝑁)

Theoremuzenom 12758 An upper integer set is denumerable. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → 𝑍 ≈ ω)

Theoremuzinf 12759 An upper integer set is infinite. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → ¬ 𝑍 ∈ Fin)

Theoremnnnfi 12760 The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
¬ ℕ ∈ Fin

Theoremuzrdgxfr 12761* 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 → (𝐺𝑁) = ((𝐻𝑁) + (𝐴𝐵)))

Theoremfzennn 12762 The cardinality of a finite set of sequential integers. (See om2uz0i 12741 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) ≈ (𝐺𝑁))

Theoremfzen2 12763 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 ∈ (ℤ𝑀) → (𝑀...𝑁) ≈ (𝐺‘((𝑁 + 1) − 𝑀)))

Theoremcardfz 12764 The cardinality of a finite set of sequential integers. (See om2uz0i 12741 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)) = (𝐺𝑁))

Theoremhashgf1o 12765 𝐺 maps ω one-to-one onto 0. (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

Theoremfzfi 12766 A finite interval of integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)
(𝑀...𝑁) ∈ Fin

Theoremfzfid 12767 Commonly used special case of fzfi 12766. (Contributed by Mario Carneiro, 25-May-2014.)
(𝜑 → (𝑀...𝑁) ∈ Fin)

Theoremfzofi 12768 Half-open integer sets are finite. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑀..^𝑁) ∈ Fin

Theoremfsequb 12769* 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)(𝐹𝑘) ∈ ℝ → ∃𝑥 ∈ ℝ ∀𝑘 ∈ (𝑀...𝑁)(𝐹𝑘) < 𝑥)

Theoremfsequb2 12770* 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 𝑦𝑥)

Theoremfseqsupcl 12771 The values of a finite real sequence have a supremum. (Contributed by NM, 20-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
((𝑁 ∈ (ℤ𝑀) ∧ 𝐹:(𝑀...𝑁)⟶ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)

Theoremfseqsupubi 12772 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) → (𝐹𝐾) ≤ sup(ran 𝐹, ℝ, < ))

Theoremnn0ennn 12773 The nonnegative integers are equinumerous to the positive integers. (Contributed by NM, 19-Jul-2004.)
0 ≈ ℕ

Theoremnnenom 12774 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

Theoremnnct 12775 is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
ℕ ≼ ω

Theoremuzindi 12776* Indirect strong induction on the upper integers. (Contributed by Stefan O'Rear, 25-Aug-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ (ℤ𝐿))    &   ((𝜑𝑅 ∈ (𝐿...𝑇) ∧ ∀𝑦(𝑆 ∈ (𝐿..^𝑅) → 𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)

Theoremaxdc4uzlem 12777* Lemma for axdc4uz 12778. (Contributed by Mario Carneiro, 8-Jan-2014.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝐴 ∈ V    &   𝐺 = (rec((𝑦 ∈ V ↦ (𝑦 + 1)), 𝑀) ↾ ω)    &   𝐻 = (𝑛 ∈ ω, 𝑥𝐴 ↦ ((𝐺𝑛)𝐹𝑥))       ((𝐶𝐴𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍𝐴 ∧ (𝑔𝑀) = 𝐶 ∧ ∀𝑘𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremaxdc4uz 12778* A version of axdc4 9275 that works on an upper set of integers instead of ω. (Contributed by Mario Carneiro, 8-Jan-2014.)
𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)       ((𝐴𝑉𝐶𝐴𝐹:(𝑍 × 𝐴)⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:𝑍𝐴 ∧ (𝑔𝑀) = 𝐶 ∧ ∀𝑘𝑍 (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔𝑘))))

Theoremssnn0fi 12779* 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 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥𝑆)))

Theoremrabssnn0fi 12780* 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 ∈ ℕ0𝜑} ∈ Fin ↔ ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ¬ 𝜑))

5.6.4  Strong induction over upper sets of integers

Theoremuzsinds 12781* Strong (or "total") induction principle over an upper set of integers. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ (ℤ𝑀) → (∀𝑦 ∈ (𝑀...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ (ℤ𝑀) → 𝜒)

Theoremnnsinds 12782* Strong (or "total") induction principle over the naturals. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ ℕ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ ℕ → 𝜒)

Theoremnn0sinds 12783* Strong (or "total") induction principle over the nonnegative integers. (Contributed by Scott Fenton, 16-May-2014.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑁 → (𝜑𝜒))    &   (𝑥 ∈ ℕ0 → (∀𝑦 ∈ (0...(𝑥 − 1))𝜓𝜑))       (𝑁 ∈ ℕ0𝜒)

5.6.5  Finitely supported functions over the nonnegative integers

Theoremfsuppmapnn0fiublem 12784* Lemma for fsuppmapnn0fiub 12785 and fsuppmapnn0fiubex 12787. (Contributed by AV, 2-Oct-2019.)
𝑈 = 𝑓𝑀 (𝑓 supp 𝑍)    &   𝑆 = sup(𝑈, ℝ, < )       ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍𝑈 ≠ ∅) → 𝑆 ∈ ℕ0))

Theoremfsuppmapnn0fiub 12785* 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 = 𝑓𝑀 (𝑓 supp 𝑍)    &   𝑆 = sup(𝑈, ℝ, < )       ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍𝑈 ≠ ∅) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆)))

Theoremfsuppmapnn0fiubOLD 12786* Obsolete proof of fsuppmapnn0fiub 12785 as of 2-Aug-2021. (Contributed by AV, 2-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑈 = 𝑓𝑀 (𝑓 supp 𝑍)    &   𝑆 = sup(𝑈, ℝ, < )       ((𝑀 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → ((∀𝑓𝑀 𝑓 finSupp 𝑍𝑈 ≠ ∅) → ∀𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑆)))

Theoremfsuppmapnn0fiubex 12787* 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 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀 (𝑓 supp 𝑍) ⊆ (0...𝑚)))

Theoremfsuppmapnn0fiub0 12788* 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 ⊆ (𝑅𝑚0) ∧ 𝑀 ∈ Fin ∧ 𝑍𝑉) → (∀𝑓𝑀 𝑓 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑓𝑀𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝑓𝑥) = 𝑍)))

Theoremsuppssfz 12789* 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.)
(𝜑𝑍𝑉)    &   (𝜑𝐹 ∈ (𝐵𝑚0))    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → (𝐹𝑥) = 𝑍))       (𝜑 → (𝐹 supp 𝑍) ⊆ (0...𝑆))

Theoremfsuppmapnn0ub 12790* 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 ∈ (𝑅𝑚0) ∧ 𝑍𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0𝑥 ∈ ℕ0 (𝑚 < 𝑥 → (𝐹𝑥) = 𝑍)))

Theoremfsuppmapnn0fz 12791* 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 ∈ (𝑅𝑚0) ∧ 𝑍𝑉) → (𝐹 finSupp 𝑍 → ∃𝑚 ∈ ℕ0 (𝐹 supp 𝑍) ⊆ (0...𝑚)))

Theoremmptnn0fsupp 12792* A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥 / 𝑘𝐶 = 0 ))       (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )

Theoremmptnn0fsuppd 12793* A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 2-Dec-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝑘 = 𝑥𝐶 = 𝐷)    &   (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝐷 = 0 ))       (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )

Theoremmptnn0fsuppr 12794* A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 23-Dec-2019.)
(𝜑0𝑉)    &   ((𝜑𝑘 ∈ ℕ0) → 𝐶𝐵)    &   (𝜑 → (𝑘 ∈ ℕ0𝐶) finSupp 0 )       (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥𝑥 / 𝑘𝐶 = 0 ))

Theoremf13idfv 12795 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:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ((𝐹‘0) ≠ (𝐹‘1) ∧ (𝐹‘0) ≠ (𝐹‘2) ∧ (𝐹‘1) ≠ (𝐹‘2))))

5.6.6  The infinite sequence builder "seq" - extension

Syntaxcseq 12796 Extend class notation with recursive sequence builder.
class seq𝑀( + , 𝐹)

Definitiondf-seq 12797* Define a general-purpose operation that builds a recursive sequence (i.e. a function on the positive integers or some other upper integer set) 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 12809 and seqp1 12811. 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 14277), by climdm 14279 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, (𝐹𝑀)⟩, 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 12741 through om2uzf1oi 12747, originally proved by Raph Levien for use with df-exp 12856 and later generalized for arbitrary recursive sequences. Definition df-sum 14411 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, (𝐹𝑀)⟩) “ ω)

Theoremseqex 12798 Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
seq𝑀( + , 𝐹) ∈ V

Theoremseqeq1 12799 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
(𝑀 = 𝑁 → seq𝑀( + , 𝐹) = seq𝑁( + , 𝐹))

Theoremseqeq2 12800 Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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