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Theorem List for Intuitionistic Logic Explorer - 11901-12000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhalfleoddlt 11901 An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁 ∧ 𝑀 ∈ β„€) β†’ ((𝑁 / 2) ≀ 𝑀 ↔ (𝑁 / 2) < 𝑀))
 
Theoremopoe 11902 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐴) ∧ (𝐡 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐡)) β†’ 2 βˆ₯ (𝐴 + 𝐡))
 
Theoremomoe 11903 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐴) ∧ (𝐡 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐡)) β†’ 2 βˆ₯ (𝐴 βˆ’ 𝐡))
 
Theoremopeo 11904 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐴) ∧ (𝐡 ∈ β„€ ∧ 2 βˆ₯ 𝐡)) β†’ Β¬ 2 βˆ₯ (𝐴 + 𝐡))
 
Theoremomeo 11905 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(((𝐴 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝐴) ∧ (𝐡 ∈ β„€ ∧ 2 βˆ₯ 𝐡)) β†’ Β¬ 2 βˆ₯ (𝐴 βˆ’ 𝐡))
 
Theoremm1expe 11906 Exponentiation of -1 by an even power. Variant of m1expeven 10569. (Contributed by AV, 25-Jun-2021.)
(2 βˆ₯ 𝑁 β†’ (-1↑𝑁) = 1)
 
Theoremm1expo 11907 Exponentiation of -1 by an odd power. (Contributed by AV, 26-Jun-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁) β†’ (-1↑𝑁) = -1)
 
Theoremm1exp1 11908 Exponentiation of negative one is one iff the exponent is even. (Contributed by AV, 20-Jun-2021.)
(𝑁 ∈ β„€ β†’ ((-1↑𝑁) = 1 ↔ 2 βˆ₯ 𝑁))
 
Theoremnn0enne 11909 A positive integer is an even nonnegative integer iff it is an even positive integer. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ β„• β†’ ((𝑁 / 2) ∈ β„•0 ↔ (𝑁 / 2) ∈ β„•))
 
Theoremnn0ehalf 11910 The half of an even nonnegative integer is a nonnegative integer. (Contributed by AV, 22-Jun-2020.) (Revised by AV, 28-Jun-2021.)
((𝑁 ∈ β„•0 ∧ 2 βˆ₯ 𝑁) β†’ (𝑁 / 2) ∈ β„•0)
 
Theoremnnehalf 11911 The half of an even positive integer is a positive integer. (Contributed by AV, 28-Jun-2021.)
((𝑁 ∈ β„• ∧ 2 βˆ₯ 𝑁) β†’ (𝑁 / 2) ∈ β„•)
 
Theoremnn0o1gt2 11912 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.)
((𝑁 ∈ β„•0 ∧ ((𝑁 + 1) / 2) ∈ β„•0) β†’ (𝑁 = 1 ∨ 2 < 𝑁))
 
Theoremnno 11913 An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.)
((𝑁 ∈ (β„€β‰₯β€˜2) ∧ ((𝑁 + 1) / 2) ∈ β„•0) β†’ ((𝑁 βˆ’ 1) / 2) ∈ β„•)
 
Theoremnn0o 11914 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)
((𝑁 ∈ β„•0 ∧ ((𝑁 + 1) / 2) ∈ β„•0) β†’ ((𝑁 βˆ’ 1) / 2) ∈ β„•0)
 
Theoremnn0ob 11915 Alternate characterizations of an odd nonnegative integer. (Contributed by AV, 4-Jun-2020.)
(𝑁 ∈ β„•0 β†’ (((𝑁 + 1) / 2) ∈ β„•0 ↔ ((𝑁 βˆ’ 1) / 2) ∈ β„•0))
 
Theoremnn0oddm1d2 11916 A positive integer is odd iff its predecessor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
(𝑁 ∈ β„•0 β†’ (Β¬ 2 βˆ₯ 𝑁 ↔ ((𝑁 βˆ’ 1) / 2) ∈ β„•0))
 
Theoremnnoddm1d2 11917 A positive integer is odd iff its successor divided by 2 is a positive integer. (Contributed by AV, 28-Jun-2021.)
(𝑁 ∈ β„• β†’ (Β¬ 2 βˆ₯ 𝑁 ↔ ((𝑁 + 1) / 2) ∈ β„•))
 
Theoremz0even 11918 0 is even. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 βˆ₯ 0
 
Theoremn2dvds1 11919 2 does not divide 1 (common case). That means 1 is odd. (Contributed by David A. Wheeler, 8-Dec-2018.)
Β¬ 2 βˆ₯ 1
 
Theoremn2dvdsm1 11920 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.)
Β¬ 2 βˆ₯ -1
 
Theoremz2even 11921 2 is even. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 23-Jun-2021.)
2 βˆ₯ 2
 
Theoremn2dvds3 11922 2 does not divide 3, i.e. 3 is an odd number. (Contributed by AV, 28-Feb-2021.)
Β¬ 2 βˆ₯ 3
 
Theoremz4even 11923 4 is an even number. (Contributed by AV, 23-Jul-2020.) (Revised by AV, 4-Jul-2021.)
2 βˆ₯ 4
 
Theorem4dvdseven 11924 An integer which is divisible by 4 is an even integer. (Contributed by AV, 4-Jul-2021.)
(4 βˆ₯ 𝑁 β†’ 2 βˆ₯ 𝑁)
 
5.1.3  The division algorithm
 
Theoremdivalglemnn 11925* Lemma for divalg 11931. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„•) β†’ βˆƒπ‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)))
 
Theoremdivalglemqt 11926 Lemma for divalg 11931. The 𝑄 = 𝑇 case involved in showing uniqueness. (Contributed by Jim Kingdon, 5-Dec-2021.)
(πœ‘ β†’ 𝐷 ∈ β„€)    &   (πœ‘ β†’ 𝑅 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ β„€)    &   (πœ‘ β†’ 𝑄 ∈ β„€)    &   (πœ‘ β†’ 𝑇 ∈ β„€)    &   (πœ‘ β†’ 𝑄 = 𝑇)    &   (πœ‘ β†’ ((𝑄 Β· 𝐷) + 𝑅) = ((𝑇 Β· 𝐷) + 𝑆))    β‡’   (πœ‘ β†’ 𝑅 = 𝑆)
 
Theoremdivalglemnqt 11927 Lemma for divalg 11931. The 𝑄 < 𝑇 case involved in showing uniqueness. (Contributed by Jim Kingdon, 4-Dec-2021.)
(πœ‘ β†’ 𝐷 ∈ β„•)    &   (πœ‘ β†’ 𝑅 ∈ β„€)    &   (πœ‘ β†’ 𝑆 ∈ β„€)    &   (πœ‘ β†’ 𝑄 ∈ β„€)    &   (πœ‘ β†’ 𝑇 ∈ β„€)    &   (πœ‘ β†’ 0 ≀ 𝑆)    &   (πœ‘ β†’ 𝑅 < 𝐷)    &   (πœ‘ β†’ ((𝑄 Β· 𝐷) + 𝑅) = ((𝑇 Β· 𝐷) + 𝑆))    β‡’   (πœ‘ β†’ Β¬ 𝑄 < 𝑇)
 
Theoremdivalglemeunn 11928* Lemma for divalg 11931. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„•) β†’ βˆƒ!π‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)))
 
Theoremdivalglemex 11929* Lemma for divalg 11931. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„€ ∧ 𝐷 β‰  0) β†’ βˆƒπ‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)))
 
Theoremdivalglemeuneg 11930* Lemma for divalg 11931. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„€ ∧ 𝐷 < 0) β†’ βˆƒ!π‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)))
 
Theoremdivalg 11931* The division algorithm (theorem). Dividing an integer 𝑁 by a nonzero integer 𝐷 produces a (unique) quotient π‘ž and a unique remainder 0 ≀ π‘Ÿ < (absβ€˜π·). Theorem 1.14 in [ApostolNT] p. 19. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„€ ∧ 𝐷 β‰  0) β†’ βˆƒ!π‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)))
 
Theoremdivalgb 11932* Express the division algorithm as stated in divalg 11931 in terms of βˆ₯. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„€ ∧ 𝐷 β‰  0) β†’ (βˆƒ!π‘Ÿ ∈ β„€ βˆƒπ‘ž ∈ β„€ (0 ≀ π‘Ÿ ∧ π‘Ÿ < (absβ€˜π·) ∧ 𝑁 = ((π‘ž Β· 𝐷) + π‘Ÿ)) ↔ βˆƒ!π‘Ÿ ∈ β„•0 (π‘Ÿ < (absβ€˜π·) ∧ 𝐷 βˆ₯ (𝑁 βˆ’ π‘Ÿ))))
 
Theoremdivalg2 11933* The division algorithm (theorem) for a positive divisor. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„•) β†’ βˆƒ!π‘Ÿ ∈ β„•0 (π‘Ÿ < 𝐷 ∧ 𝐷 βˆ₯ (𝑁 βˆ’ π‘Ÿ)))
 
Theoremdivalgmod 11934 The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor (compare divalg2 11933 and divalgb 11932). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„•) β†’ (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 ∈ β„•0 ∧ (𝑅 < 𝐷 ∧ 𝐷 βˆ₯ (𝑁 βˆ’ 𝑅)))))
 
Theoremdivalgmodcl 11935 The result of the mod operator satisfies the requirements for the remainder 𝑅 in the division algorithm for a positive divisor. Variant of divalgmod 11934. (Contributed by Stefan O'Rear, 17-Oct-2014.) (Proof shortened by AV, 21-Aug-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„• ∧ 𝑅 ∈ β„•0) β†’ (𝑅 = (𝑁 mod 𝐷) ↔ (𝑅 < 𝐷 ∧ 𝐷 βˆ₯ (𝑁 βˆ’ 𝑅))))
 
Theoremmodremain 11936* The result of the modulo operation is the remainder of the division algorithm. (Contributed by AV, 19-Aug-2021.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„• ∧ (𝑅 ∈ β„•0 ∧ 𝑅 < 𝐷)) β†’ ((𝑁 mod 𝐷) = 𝑅 ↔ βˆƒπ‘§ ∈ β„€ ((𝑧 Β· 𝐷) + 𝑅) = 𝑁))
 
Theoremndvdssub 11937 Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 βˆ’ 1, 𝑁 βˆ’ 2... 𝑁 βˆ’ (𝐷 βˆ’ 1). (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„• ∧ (𝐾 ∈ β„• ∧ 𝐾 < 𝐷)) β†’ (𝐷 βˆ₯ 𝑁 β†’ Β¬ 𝐷 βˆ₯ (𝑁 βˆ’ 𝐾)))
 
Theoremndvdsadd 11938 Corollary of the division algorithm. If an integer 𝐷 greater than 1 divides 𝑁, then it does not divide any of 𝑁 + 1, 𝑁 + 2... 𝑁 + (𝐷 βˆ’ 1). (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„• ∧ (𝐾 ∈ β„• ∧ 𝐾 < 𝐷)) β†’ (𝐷 βˆ₯ 𝑁 β†’ Β¬ 𝐷 βˆ₯ (𝑁 + 𝐾)))
 
Theoremndvdsp1 11939 Special case of ndvdsadd 11938. If an integer 𝐷 greater than 1 divides 𝑁, it does not divide 𝑁 + 1. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑁 ∈ β„€ ∧ 𝐷 ∈ β„• ∧ 1 < 𝐷) β†’ (𝐷 βˆ₯ 𝑁 β†’ Β¬ 𝐷 βˆ₯ (𝑁 + 1)))
 
Theoremndvdsi 11940 A quick test for non-divisibility. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝐴 ∈ β„•    &   π‘„ ∈ β„•0    &   π‘… ∈ β„•    &   ((𝐴 Β· 𝑄) + 𝑅) = 𝐡    &   π‘… < 𝐴    β‡’    Β¬ 𝐴 βˆ₯ 𝐡
 
Theoremflodddiv4 11941 The floor of an odd integer divided by 4. (Contributed by AV, 17-Jun-2021.)
((𝑀 ∈ β„€ ∧ 𝑁 = ((2 Β· 𝑀) + 1)) β†’ (βŒŠβ€˜(𝑁 / 4)) = if(2 βˆ₯ 𝑀, (𝑀 / 2), ((𝑀 βˆ’ 1) / 2)))
 
Theoremfldivndvdslt 11942 The floor of an integer divided by a nonzero integer not dividing the first integer is less than the integer divided by the positive integer. (Contributed by AV, 4-Jul-2021.)
((𝐾 ∈ β„€ ∧ (𝐿 ∈ β„€ ∧ 𝐿 β‰  0) ∧ Β¬ 𝐿 βˆ₯ 𝐾) β†’ (βŒŠβ€˜(𝐾 / 𝐿)) < (𝐾 / 𝐿))
 
Theoremflodddiv4lt 11943 The floor of an odd number divided by 4 is less than the odd number divided by 4. (Contributed by AV, 4-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁) β†’ (βŒŠβ€˜(𝑁 / 4)) < (𝑁 / 4))
 
Theoremflodddiv4t2lthalf 11944 The floor of an odd number divided by 4, multiplied by 2 is less than the half of the odd number. (Contributed by AV, 4-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁) β†’ ((βŒŠβ€˜(𝑁 / 4)) Β· 2) < (𝑁 / 2))
 
5.1.4  The greatest common divisor operator
 
Syntaxcgcd 11945 Extend the definition of a class to include the greatest common divisor operator.
class gcd
 
Definitiondf-gcd 11946* Define the gcd operator. For example, (-6 gcd 9) = 3 (ex-gcd 14568). (Contributed by Paul Chapman, 21-Mar-2011.)
gcd = (π‘₯ ∈ β„€, 𝑦 ∈ β„€ ↦ if((π‘₯ = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ π‘₯ ∧ 𝑛 βˆ₯ 𝑦)}, ℝ, < )))
 
Theoremgcdmndc 11947 Decidablity lemma used in various proofs related to gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ DECID (𝑀 = 0 ∧ 𝑁 = 0))
 
Theoremzsupcllemstep 11948* Lemma for zsupcl 11950. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)
((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID πœ“)    β‡’   (𝐾 ∈ (β„€β‰₯β€˜π‘€) β†’ (((πœ‘ ∧ βˆ€π‘› ∈ (β„€β‰₯β€˜πΎ) Β¬ πœ“) β†’ βˆƒπ‘₯ ∈ β„€ (βˆ€π‘¦ ∈ {𝑛 ∈ β„€ ∣ πœ“} Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ {𝑛 ∈ β„€ ∣ πœ“}𝑦 < 𝑧))) β†’ ((πœ‘ ∧ βˆ€π‘› ∈ (β„€β‰₯β€˜(𝐾 + 1)) Β¬ πœ“) β†’ βˆƒπ‘₯ ∈ β„€ (βˆ€π‘¦ ∈ {𝑛 ∈ β„€ ∣ πœ“} Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ {𝑛 ∈ β„€ ∣ πœ“}𝑦 < 𝑧)))))
 
Theoremzsupcllemex 11949* Lemma for zsupcl 11950. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (𝑛 = 𝑀 β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ πœ’)    &   ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID πœ“)    &   (πœ‘ β†’ βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘€)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘—) Β¬ πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ (βˆ€π‘¦ ∈ {𝑛 ∈ β„€ ∣ πœ“} Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ {𝑛 ∈ β„€ ∣ πœ“}𝑦 < 𝑧)))
 
Theoremzsupcl 11950* Closure of supremum for decidable integer properties. The property which defines the set we are taking the supremum of must (a) be true at 𝑀 (which corresponds to the nonempty condition of classical supremum theorems), (b) decidable at each value after 𝑀, and (c) be false after 𝑗 (which corresponds to the upper bound condition found in classical supremum theorems). (Contributed by Jim Kingdon, 7-Dec-2021.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (𝑛 = 𝑀 β†’ (πœ“ ↔ πœ’))    &   (πœ‘ β†’ πœ’)    &   ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘€)) β†’ DECID πœ“)    &   (πœ‘ β†’ βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘€)βˆ€π‘› ∈ (β„€β‰₯β€˜π‘—) Β¬ πœ“)    β‡’   (πœ‘ β†’ sup({𝑛 ∈ β„€ ∣ πœ“}, ℝ, < ) ∈ (β„€β‰₯β€˜π‘€))
 
Theoremzssinfcl 11951* The infimum of a set of integers is an element of the set. (Contributed by Jim Kingdon, 16-Jan-2022.)
(πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐡 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐡 𝑧 < 𝑦)))    &   (πœ‘ β†’ 𝐡 βŠ† β„€)    &   (πœ‘ β†’ inf(𝐡, ℝ, < ) ∈ β„€)    β‡’   (πœ‘ β†’ inf(𝐡, ℝ, < ) ∈ 𝐡)
 
Theoreminfssuzex 11952* Existence of the infimum of a subset of an upper set of integers. (Contributed by Jim Kingdon, 13-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘† = {𝑛 ∈ (β„€β‰₯β€˜π‘€) ∣ πœ“}    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   ((πœ‘ ∧ 𝑛 ∈ (𝑀...𝐴)) β†’ DECID πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝑆 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝑆 𝑧 < 𝑦)))
 
Theoreminfssuzledc 11953* The infimum of a subset of an upper set of integers is less than or equal to all members of the subset. (Contributed by Jim Kingdon, 13-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘† = {𝑛 ∈ (β„€β‰₯β€˜π‘€) ∣ πœ“}    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   ((πœ‘ ∧ 𝑛 ∈ (𝑀...𝐴)) β†’ DECID πœ“)    β‡’   (πœ‘ β†’ inf(𝑆, ℝ, < ) ≀ 𝐴)
 
Theoreminfssuzcldc 11954* The infimum of a subset of an upper set of integers belongs to the subset. (Contributed by Jim Kingdon, 20-Jan-2022.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   π‘† = {𝑛 ∈ (β„€β‰₯β€˜π‘€) ∣ πœ“}    &   (πœ‘ β†’ 𝐴 ∈ 𝑆)    &   ((πœ‘ ∧ 𝑛 ∈ (𝑀...𝐴)) β†’ DECID πœ“)    β‡’   (πœ‘ β†’ inf(𝑆, ℝ, < ) ∈ 𝑆)
 
Theoremsuprzubdc 11955* The supremum of a bounded-above decidable set of integers is greater than any member of the set. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    β‡’   (πœ‘ β†’ 𝐡 ≀ sup(𝐴, ℝ, < ))
 
Theoremnninfdcex 11956* A decidable set of natural numbers has an infimum. (Contributed by Jim Kingdon, 28-Sep-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„•)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„• DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘¦ 𝑦 ∈ 𝐴)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))
 
Theoremzsupssdc 11957* An inhabited decidable bounded subset of integers has a supremum in the set. (The proof does not use ax-pre-suploc 7934.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 5-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ 𝐡 (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
Theoremsuprzcl2dc 11958* The supremum of a bounded-above decidable set of integers is a member of the set. (This theorem avoids ax-pre-suploc 7934.) (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by Jim Kingdon, 6-Oct-2024.)
(πœ‘ β†’ 𝐴 βŠ† β„€)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ β„€ DECID π‘₯ ∈ 𝐴)    &   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯)    &   (πœ‘ β†’ βˆƒπ‘₯ π‘₯ ∈ 𝐴)    β‡’   (πœ‘ β†’ sup(𝐴, ℝ, < ) ∈ 𝐴)
 
Theoremdvdsbnd 11959* There is an upper bound to the divisors of a nonzero integer. (Contributed by Jim Kingdon, 11-Dec-2021.)
((𝐴 ∈ β„€ ∧ 𝐴 β‰  0) β†’ βˆƒπ‘› ∈ β„• βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›) Β¬ π‘š βˆ₯ 𝐴)
 
Theoremgcdsupex 11960* Existence of the supremum used in defining gcd. (Contributed by Jim Kingdon, 12-Dec-2021.)
(((𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€) ∧ Β¬ (𝑋 = 0 ∧ π‘Œ = 0)) β†’ βˆƒπ‘₯ ∈ β„€ (βˆ€π‘¦ ∈ {𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ 𝑋 ∧ 𝑛 βˆ₯ π‘Œ)} Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ {𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ 𝑋 ∧ 𝑛 βˆ₯ π‘Œ)}𝑦 < 𝑧)))
 
Theoremgcdsupcl 11961* Closure of the supremum used in defining gcd. A lemma for gcdval 11962 and gcdn0cl 11965. (Contributed by Jim Kingdon, 11-Dec-2021.)
(((𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€) ∧ Β¬ (𝑋 = 0 ∧ π‘Œ = 0)) β†’ sup({𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ 𝑋 ∧ 𝑛 βˆ₯ π‘Œ)}, ℝ, < ) ∈ β„•)
 
Theoremgcdval 11962* The value of the gcd operator. (𝑀 gcd 𝑁) is the greatest common divisor of 𝑀 and 𝑁. If 𝑀 and 𝑁 are both 0, the result is defined conventionally as 0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd 𝑁) = if((𝑀 = 0 ∧ 𝑁 = 0), 0, sup({𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ 𝑀 ∧ 𝑛 βˆ₯ 𝑁)}, ℝ, < )))
 
Theoremgcd0val 11963 The value, by convention, of the gcd operator when both operands are 0. (Contributed by Paul Chapman, 21-Mar-2011.)
(0 gcd 0) = 0
 
Theoremgcdn0val 11964* The value of the gcd operator when at least one operand is nonzero. (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ Β¬ (𝑀 = 0 ∧ 𝑁 = 0)) β†’ (𝑀 gcd 𝑁) = sup({𝑛 ∈ β„€ ∣ (𝑛 βˆ₯ 𝑀 ∧ 𝑛 βˆ₯ 𝑁)}, ℝ, < ))
 
Theoremgcdn0cl 11965 Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ Β¬ (𝑀 = 0 ∧ 𝑁 = 0)) β†’ (𝑀 gcd 𝑁) ∈ β„•)
 
Theoremgcddvds 11966 The gcd of two integers divides each of them. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((𝑀 gcd 𝑁) βˆ₯ 𝑀 ∧ (𝑀 gcd 𝑁) βˆ₯ 𝑁))
 
Theoremdvdslegcd 11967 An integer which divides both operands of the gcd operator is bounded by it. (Contributed by Paul Chapman, 21-Mar-2011.)
(((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ Β¬ (𝑀 = 0 ∧ 𝑁 = 0)) β†’ ((𝐾 βˆ₯ 𝑀 ∧ 𝐾 βˆ₯ 𝑁) β†’ 𝐾 ≀ (𝑀 gcd 𝑁)))
 
Theoremnndvdslegcd 11968 A positive integer which divides both positive operands of the gcd operator is bounded by it. (Contributed by AV, 9-Aug-2020.)
((𝐾 ∈ β„• ∧ 𝑀 ∈ β„• ∧ 𝑁 ∈ β„•) β†’ ((𝐾 βˆ₯ 𝑀 ∧ 𝐾 βˆ₯ 𝑁) β†’ 𝐾 ≀ (𝑀 gcd 𝑁)))
 
Theoremgcdcl 11969 Closure of the gcd operator. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd 𝑁) ∈ β„•0)
 
Theoremgcdnncl 11970 Closure of the gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
((𝑀 ∈ β„• ∧ 𝑁 ∈ β„•) β†’ (𝑀 gcd 𝑁) ∈ β„•)
 
Theoremgcdcld 11971 Closure of the gcd operator. (Contributed by Mario Carneiro, 29-May-2016.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) ∈ β„•0)
 
Theoremgcd2n0cl 11972 Closure of the gcd operator if the second operand is not 0. (Contributed by AV, 10-Jul-2021.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 𝑁 β‰  0) β†’ (𝑀 gcd 𝑁) ∈ β„•)
 
Theoremzeqzmulgcd 11973* An integer is the product of an integer and the gcd of it and another integer. (Contributed by AV, 11-Jul-2021.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ βˆƒπ‘› ∈ β„€ 𝐴 = (𝑛 Β· (𝐴 gcd 𝐡)))
 
Theoremdivgcdz 11974 An integer divided by the gcd of it and a nonzero integer is an integer. (Contributed by AV, 11-Jul-2021.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝐡 β‰  0) β†’ (𝐴 / (𝐴 gcd 𝐡)) ∈ β„€)
 
Theoremgcdf 11975 Domain and codomain of the gcd operator. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 16-Nov-2013.)
gcd :(β„€ Γ— β„€)βŸΆβ„•0
 
Theoremgcdcom 11976 The gcd operator is commutative. Theorem 1.4(a) in [ApostolNT] p. 16. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
Theoremgcdcomd 11977 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
Theoremdivgcdnn 11978 A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„€) β†’ (𝐴 / (𝐴 gcd 𝐡)) ∈ β„•)
 
Theoremdivgcdnnr 11979 A positive integer divided by the gcd of it and another integer is a positive integer. (Contributed by AV, 10-Jul-2021.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„€) β†’ (𝐴 / (𝐡 gcd 𝐴)) ∈ β„•)
 
Theoremgcdeq0 11980 The gcd of two integers is zero iff they are both zero. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((𝑀 gcd 𝑁) = 0 ↔ (𝑀 = 0 ∧ 𝑁 = 0)))
 
Theoremgcdn0gt0 11981 The gcd of two integers is positive (nonzero) iff they are not both zero. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (Β¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ 0 < (𝑀 gcd 𝑁)))
 
Theoremgcd0id 11982 The gcd of 0 and an integer is the integer's absolute value. (Contributed by Paul Chapman, 21-Mar-2011.)
(𝑁 ∈ β„€ β†’ (0 gcd 𝑁) = (absβ€˜π‘))
 
Theoremgcdid0 11983 The gcd of an integer and 0 is the integer's absolute value. Theorem 1.4(d)2 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 31-Mar-2011.)
(𝑁 ∈ β„€ β†’ (𝑁 gcd 0) = (absβ€˜π‘))
 
Theoremnn0gcdid0 11984 The gcd of a nonnegative integer with 0 is itself. (Contributed by Paul Chapman, 31-Mar-2011.)
(𝑁 ∈ β„•0 β†’ (𝑁 gcd 0) = 𝑁)
 
Theoremgcdneg 11985 Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁))
 
Theoremneggcd 11986 Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁))
 
Theoremgcdaddm 11987 Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd 𝑁) = (𝑀 gcd (𝑁 + (𝐾 Β· 𝑀))))
 
Theoremgcdadd 11988 The GCD of two numbers is the same as the GCD of the left and their sum. (Contributed by Scott Fenton, 20-Apr-2014.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 gcd 𝑁) = (𝑀 gcd (𝑁 + 𝑀)))
 
Theoremgcdid 11989 The gcd of a number and itself is its absolute value. (Contributed by Paul Chapman, 31-Mar-2011.)
(𝑁 ∈ β„€ β†’ (𝑁 gcd 𝑁) = (absβ€˜π‘))
 
Theoremgcd1 11990 The gcd of a number with 1 is 1. Theorem 1.4(d)1 in [ApostolNT] p. 16. (Contributed by Mario Carneiro, 19-Feb-2014.)
(𝑀 ∈ β„€ β†’ (𝑀 gcd 1) = 1)
 
Theoremgcdabs 11991 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((absβ€˜π‘€) gcd (absβ€˜π‘)) = (𝑀 gcd 𝑁))
 
Theoremgcdabs1 11992 gcd of the absolute value of the first operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ ((absβ€˜π‘) gcd 𝑀) = (𝑁 gcd 𝑀))
 
Theoremgcdabs2 11993 gcd of the absolute value of the second operator. (Contributed by Scott Fenton, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ (𝑁 gcd (absβ€˜π‘€)) = (𝑁 gcd 𝑀))
 
Theoremmodgcd 11994 The gcd remains unchanged if one operand is replaced with its remainder modulo the other. (Contributed by Paul Chapman, 31-Mar-2011.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„•) β†’ ((𝑀 mod 𝑁) gcd 𝑁) = (𝑀 gcd 𝑁))
 
Theorem1gcd 11995 The GCD of one and an integer is one. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝑀 ∈ β„€ β†’ (1 gcd 𝑀) = 1)
 
Theoremgcdmultipled 11996 The greatest common divisor of a nonnegative integer 𝑀 and a multiple of it is 𝑀 itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝑀 ∈ β„•0)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd (𝑁 Β· 𝑀)) = 𝑀)
 
Theoremdvdsgcdidd 11997 The greatest common divisor of a positive integer and another integer it divides is itself. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) = 𝑀)
 
Theorem6gcd4e2 11998 The greatest common divisor of six and four is two. To calculate this gcd, a simple form of Euclid's algorithm is used: (6 gcd 4) = ((4 + 2) gcd 4) = (2 gcd 4) and (2 gcd 4) = (2 gcd (2 + 2)) = (2 gcd 2) = 2. (Contributed by AV, 27-Aug-2020.)
(6 gcd 4) = 2
 
5.1.5  BΓ©zout's identity
 
Theorembezoutlemnewy 11999* Lemma for BΓ©zout's identity. The is-bezout predicate holds for (𝑦 mod π‘Š). (Contributed by Jim Kingdon, 6-Jan-2022.)
(πœ‘ ↔ βˆƒπ‘  ∈ β„€ βˆƒπ‘‘ ∈ β„€ π‘Ÿ = ((𝐴 Β· 𝑠) + (𝐡 Β· 𝑑)))    &   (πœƒ β†’ 𝐴 ∈ β„•0)    &   (πœƒ β†’ 𝐡 ∈ β„•0)    &   (πœƒ β†’ π‘Š ∈ β„•)    &   (πœƒ β†’ [𝑦 / π‘Ÿ]πœ‘)    &   (πœƒ β†’ 𝑦 ∈ β„•0)    &   (πœƒ β†’ [π‘Š / π‘Ÿ]πœ‘)    β‡’   (πœƒ β†’ [(𝑦 mod π‘Š) / π‘Ÿ]πœ‘)
 
Theorembezoutlemstep 12000* Lemma for BΓ©zout's identity. This is the induction step for the proof by induction. (Contributed by Jim Kingdon, 3-Jan-2022.)
(πœ‘ ↔ βˆƒπ‘  ∈ β„€ βˆƒπ‘‘ ∈ β„€ π‘Ÿ = ((𝐴 Β· 𝑠) + (𝐡 Β· 𝑑)))    &   (πœƒ β†’ 𝐴 ∈ β„•0)    &   (πœƒ β†’ 𝐡 ∈ β„•0)    &   (πœƒ β†’ π‘Š ∈ β„•)    &   (πœƒ β†’ [𝑦 / π‘Ÿ]πœ‘)    &   (πœƒ β†’ 𝑦 ∈ β„•0)    &   (πœƒ β†’ [π‘Š / π‘Ÿ]πœ‘)    &   (πœ“ ↔ βˆ€π‘§ ∈ β„•0 (𝑧 βˆ₯ π‘Ÿ β†’ (𝑧 βˆ₯ π‘₯ ∧ 𝑧 βˆ₯ 𝑦)))    &   ((πœƒ ∧ [(𝑦 mod π‘Š) / π‘Ÿ]πœ‘) β†’ βˆƒπ‘Ÿ ∈ β„•0 ([(𝑦 mod π‘Š) / π‘₯][π‘Š / 𝑦]πœ“ ∧ πœ‘))    &   β„²π‘₯πœƒ    &   β„²π‘Ÿπœƒ    β‡’   (πœƒ β†’ βˆƒπ‘Ÿ ∈ β„•0 ([π‘Š / π‘₯]πœ“ ∧ πœ‘))
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