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Theorem List for Intuitionistic Logic Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremexpcl2lemap 8901* Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)
𝐹 ⊆ ℂ    &   ((x 𝐹 y 𝐹) → (x · y) 𝐹)    &   1 𝐹    &   ((x 𝐹 x # 0) → (1 / x) 𝐹)       ((A 𝐹 A # 0 B ℤ) → (AB) 𝐹)

Theoremnnexpcl 8902 Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.)
((A 𝑁 0) → (A𝑁) ℕ)

Theoremnn0expcl 8903 Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005.)
((A 0 𝑁 0) → (A𝑁) 0)

Theoremzexpcl 8904 Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005.)
((A 𝑁 0) → (A𝑁) ℤ)

Theoremqexpcl 8905 Closure of exponentiation of rationals. (Contributed by NM, 16-Dec-2005.)
((A 𝑁 0) → (A𝑁) ℚ)

Theoremreexpcl 8906 Closure of exponentiation of reals. (Contributed by NM, 14-Dec-2005.)
((A 𝑁 0) → (A𝑁) ℝ)

Theoremexpcl 8907 Closure law for nonnegative integer exponentiation. (Contributed by NM, 26-May-2005.)
((A 𝑁 0) → (A𝑁) ℂ)

Theoremrpexpcl 8908 Closure law for exponentiation of positive reals. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 9-Sep-2014.)
((A + 𝑁 ℤ) → (A𝑁) +)

Theoremreexpclzap 8909 Closure of exponentiation of reals. (Contributed by Jim Kingdon, 9-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A𝑁) ℝ)

Theoremqexpclz 8910 Closure of exponentiation of rational numbers. (Contributed by Mario Carneiro, 9-Sep-2014.)
((A A ≠ 0 𝑁 ℤ) → (A𝑁) ℚ)

Theoremm1expcl2 8911 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ℤ → (-1↑𝑁) {-1, 1})

Theoremm1expcl 8912 Closure of exponentiation of negative one. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑁 ℤ → (-1↑𝑁) ℤ)

Theoremexpclzaplem 8913* Closure law for integer exponentiation. Lemma for expclzap 8914 and expap0i 8921. (Contributed by Jim Kingdon, 9-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A𝑁) {z ℂ ∣ z # 0})

Theoremexpclzap 8914 Closure law for integer exponentiation. (Contributed by Jim Kingdon, 9-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A𝑁) ℂ)

Theoremnn0expcli 8915 Closure of exponentiation of nonnegative integers. (Contributed by Mario Carneiro, 17-Apr-2015.)
A 0    &   𝑁 0       (A𝑁) 0

Theoremnn0sqcl 8916 The square of a nonnegative integer is a nonnegative integer. (Contributed by Stefan O'Rear, 16-Oct-2014.)
(A 0 → (A↑2) 0)

Theoremexpm1t 8917 Exponentiation in terms of predecessor exponent. (Contributed by NM, 19-Dec-2005.)
((A 𝑁 ℕ) → (A𝑁) = ((A↑(𝑁 − 1)) · A))

Theorem1exp 8918 Value of one raised to a nonnegative integer power. (Contributed by NM, 15-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝑁 ℤ → (1↑𝑁) = 1)

Theoremexpap0 8919 Positive integer exponentiation is apart from zero iff its mantissa is apart from zero. That it is easier to prove this first, and then prove expeq0 8920 in terms of it, rather than the other way around, is perhaps an illustration of the maxim "In constructive analysis, the apartness is more basic [ than ] equality." ([Geuvers], p. 1). (Contributed by Jim Kingdon, 10-Jun-2020.)
((A 𝑁 ℕ) → ((A𝑁) # 0 ↔ A # 0))

Theoremexpeq0 8920 Positive integer exponentiation is 0 iff its mantissa is 0. (Contributed by NM, 23-Feb-2005.)
((A 𝑁 ℕ) → ((A𝑁) = 0 ↔ A = 0))

Theoremexpap0i 8921 Integer exponentiation is apart from zero if its mantissa is apart from zero. (Contributed by Jim Kingdon, 10-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A𝑁) # 0)

Theoremexpgt0 8922 Nonnegative integer exponentiation with a positive mantissa is positive. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((A 𝑁 0 < A) → 0 < (A𝑁))

Theoremexpnegzap 8923 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 4-Jun-2014.)
((A A # 0 𝑁 ℤ) → (A↑-𝑁) = (1 / (A𝑁)))

Theorem0exp 8924 Value of zero raised to a positive integer power. (Contributed by NM, 19-Aug-2004.)
(𝑁 ℕ → (0↑𝑁) = 0)

Theoremexpge0 8925 Nonnegative integer exponentiation with a nonnegative mantissa is nonnegative. (Contributed by NM, 16-Dec-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((A 𝑁 0 0 ≤ A) → 0 ≤ (A𝑁))

Theoremexpge1 8926 Nonnegative integer exponentiation with a mantissa greater than or equal to 1 is greater than or equal to 1. (Contributed by NM, 21-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((A 𝑁 0 1 ≤ A) → 1 ≤ (A𝑁))

Theoremexpgt1 8927 Positive integer exponentiation with a mantissa greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
((A 𝑁 1 < A) → 1 < (A𝑁))

Theoremmulexp 8928 Positive 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 B 𝑁 0) → ((A · B)↑𝑁) = ((A𝑁) · (B𝑁)))

Theoremmulexpzap 8929 Integer exponentiation of a product. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((A A # 0) (B B # 0) 𝑁 ℤ) → ((A · B)↑𝑁) = ((A𝑁) · (B𝑁)))

Theoremexprecap 8930 Nonnegative integer exponentiation of a reciprocal. (Contributed by Jim Kingdon, 10-Jun-2020.)
((A A # 0 𝑁 ℤ) → ((1 / A)↑𝑁) = (1 / (A𝑁)))

Theoremexpadd 8931 Sum of exponents law for nonnegative integer exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by NM, 30-Nov-2004.)
((A 𝑀 0 𝑁 0) → (A↑(𝑀 + 𝑁)) = ((A𝑀) · (A𝑁)))

(((A A # 0) (𝑀 -𝑀 ℕ) 𝑁 0) → (A↑(𝑀 + 𝑁)) = ((A𝑀) · (A𝑁)))

Theoremexpaddzap 8933 Sum of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 10-Jun-2020.)
(((A A # 0) (𝑀 𝑁 ℤ)) → (A↑(𝑀 + 𝑁)) = ((A𝑀) · (A𝑁)))

Theoremexpmul 8934 Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135, restricted to nonnegative integer exponents. (Contributed by NM, 4-Jan-2006.)
((A 𝑀 0 𝑁 0) → (A↑(𝑀 · 𝑁)) = ((A𝑀)↑𝑁))

Theoremexpmulzap 8935 Product of exponents law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((A A # 0) (𝑀 𝑁 ℤ)) → (A↑(𝑀 · 𝑁)) = ((A𝑀)↑𝑁))

Theoremexpsubap 8936 Exponent subtraction law for nonnegative integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
(((A A # 0) (𝑀 𝑁 ℤ)) → (A↑(𝑀𝑁)) = ((A𝑀) / (A𝑁)))

Theoremexpp1zap 8937 Value of a nonzero complex number raised to an integer power plus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A↑(𝑁 + 1)) = ((A𝑁) · A))

Theoremexpm1ap 8938 Value of a complex number raised to an integer power minus one. (Contributed by Jim Kingdon, 11-Jun-2020.)
((A A # 0 𝑁 ℤ) → (A↑(𝑁 − 1)) = ((A𝑁) / A))

Theoremexpdivap 8939 Nonnegative integer exponentiation of a quotient. (Contributed by Jim Kingdon, 11-Jun-2020.)
((A (B B # 0) 𝑁 0) → ((A / B)↑𝑁) = ((A𝑁) / (B𝑁)))

Theoremltexp2a 8940 Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
(((A 𝑀 𝑁 ℤ) (1 < A 𝑀 < 𝑁)) → (A𝑀) < (A𝑁))

Theoremleexp2a 8941 Weak ordering relationship for exponentiation. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 5-Jun-2014.)
((A 1 ≤ A 𝑁 (ℤ𝑀)) → (A𝑀) ≤ (A𝑁))

Theoremleexp2r 8942 Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)
(((A 𝑀 0 𝑁 (ℤ𝑀)) (0 ≤ A A ≤ 1)) → (A𝑁) ≤ (A𝑀))

Theoremleexp1a 8943 Weak mantissa ordering relationship for exponentiation. (Contributed by NM, 18-Dec-2005.)
(((A B 𝑁 0) (0 ≤ A AB)) → (A𝑁) ≤ (B𝑁))

Theoremexple1 8944 Nonnegative integer exponentiation with a mantissa between 0 and 1 inclusive is less than or equal to 1. (Contributed by Paul Chapman, 29-Dec-2007.) (Revised by Mario Carneiro, 5-Jun-2014.)
(((A 0 ≤ A A ≤ 1) 𝑁 0) → (A𝑁) ≤ 1)

Theoremexpubnd 8945 An upper bound on A𝑁 when 2 ≤ A. (Contributed by NM, 19-Dec-2005.)
((A 𝑁 0 2 ≤ A) → (A𝑁) ≤ ((2↑𝑁) · ((A − 1)↑𝑁)))

Theoremsqval 8946 Value of the square of a complex number. (Contributed by Raph Levien, 10-Apr-2004.)
(A ℂ → (A↑2) = (A · A))

Theoremsqneg 8947 The square of the negative of a number.) (Contributed by NM, 15-Jan-2006.)
(A ℂ → (-A↑2) = (A↑2))

Theoremsqsubswap 8948 Swap the order of subtraction in a square. (Contributed by Scott Fenton, 10-Jun-2013.)
((A B ℂ) → ((AB)↑2) = ((BA)↑2))

Theoremsqcl 8949 Closure of square. (Contributed by NM, 10-Aug-1999.)
(A ℂ → (A↑2) ℂ)

Theoremsqmul 8950 Distribution of square over multiplication. (Contributed by NM, 21-Mar-2008.)
((A B ℂ) → ((A · B)↑2) = ((A↑2) · (B↑2)))

Theoremsqeq0 8951 A number is zero iff its square is zero. (Contributed by NM, 11-Mar-2006.)
(A ℂ → ((A↑2) = 0 ↔ A = 0))

Theoremsqdivap 8952 Distribution of square over division. (Contributed by Jim Kingdon, 11-Jun-2020.)
((A B B # 0) → ((A / B)↑2) = ((A↑2) / (B↑2)))

Theoremsqne0 8953 A number is nonzero iff its square is nonzero. (Contributed by NM, 11-Mar-2006.)
(A ℂ → ((A↑2) ≠ 0 ↔ A ≠ 0))

Theoremresqcl 8954 Closure of the square of a real number. (Contributed by NM, 18-Oct-1999.)
(A ℝ → (A↑2) ℝ)

Theoremsqgt0ap 8955 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 11-Jun-2020.)
((A A # 0) → 0 < (A↑2))

Theoremnnsqcl 8956 The naturals are closed under squaring. (Contributed by Scott Fenton, 29-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(A ℕ → (A↑2) ℕ)

Theoremzsqcl 8957 Integers are closed under squaring. (Contributed by Scott Fenton, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(A ℤ → (A↑2) ℤ)

Theoremqsqcl 8958 The square of a rational is rational. (Contributed by Stefan O'Rear, 15-Sep-2014.)
(A ℚ → (A↑2) ℚ)

Theoremsq11 8959 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 0 ≤ A) (B 0 ≤ B)) → ((A↑2) = (B↑2) ↔ A = B))

Theoremlt2sq 8960 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006.)
(((A 0 ≤ A) (B 0 ≤ B)) → (A < B ↔ (A↑2) < (B↑2)))

Theoremle2sq 8961 The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999.)
(((A 0 ≤ A) (B 0 ≤ B)) → (AB ↔ (A↑2) ≤ (B↑2)))

Theoremle2sq2 8962 The square of a 'less than or equal to' ordering. (Contributed by NM, 21-Mar-2008.)
(((A 0 ≤ A) (B AB)) → (A↑2) ≤ (B↑2))

Theoremsqge0 8963 A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999.)
(A ℝ → 0 ≤ (A↑2))

Theoremzsqcl2 8964 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) 0)

Theoremsumsqeq0 8965 Two real numbers are equal to 0 iff their Euclidean norm is. (Contributed by NM, 29-Apr-2005.) (Revised by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 28-May-2016.)
((A B ℝ) → ((A = 0 B = 0) ↔ ((A↑2) + (B↑2)) = 0))

Theoremsqvali 8966 Value of square. Inference version. (Contributed by NM, 1-Aug-1999.)
A        (A↑2) = (A · A)

Theoremsqcli 8967 Closure of square. (Contributed by NM, 2-Aug-1999.)
A        (A↑2)

Theoremsqeq0i 8968 A number is zero iff its square is zero. (Contributed by NM, 2-Oct-1999.)
A        ((A↑2) = 0 ↔ A = 0)

Theoremsqmuli 8969 Distribution of square over multiplication. (Contributed by NM, 3-Sep-1999.)
A     &   B        ((A · B)↑2) = ((A↑2) · (B↑2))

Theoremsqdivapi 8970 Distribution of square over division. (Contributed by Jim Kingdon, 12-Jun-2020.)
A     &   B     &   B # 0       ((A / B)↑2) = ((A↑2) / (B↑2))

Theoremresqcli 8971 Closure of square in reals. (Contributed by NM, 2-Aug-1999.)
A        (A↑2)

Theoremsqgt0api 8972 The square of a nonzero real is positive. (Contributed by Jim Kingdon, 12-Jun-2020.)
A        (A # 0 → 0 < (A↑2))

Theoremsqge0i 8973 A square of a real is nonnegative. (Contributed by NM, 3-Aug-1999.)
A        0 ≤ (A↑2)

Theoremlt2sqi 8974 The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 12-Sep-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → (A < B ↔ (A↑2) < (B↑2)))

Theoremle2sqi 8975 The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → (AB ↔ (A↑2) ≤ (B↑2)))

Theoremsq11i 8976 The square function is one-to-one for nonnegative reals. (Contributed by NM, 27-Oct-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → ((A↑2) = (B↑2) ↔ A = B))

Theoremsq0 8977 The square of 0 is 0. (Contributed by NM, 6-Jun-2006.)
(0↑2) = 0

Theoremsq0i 8978 If a number is zero, its square is zero. (Contributed by FL, 10-Dec-2006.)
(A = 0 → (A↑2) = 0)

Theoremsq0id 8979 If a number is zero, its square is zero. Deduction form of sq0i 8978. Converse of sqeq0d 9013. (Contributed by David Moews, 28-Feb-2017.)
(φA = 0)       (φ → (A↑2) = 0)

Theoremsq1 8980 The square of 1 is 1. (Contributed by NM, 22-Aug-1999.)
(1↑2) = 1

Theoremneg1sqe1 8981 -1 squared is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(-1↑2) = 1

Theoremsq2 8982 The square of 2 is 4. (Contributed by NM, 22-Aug-1999.)
(2↑2) = 4

Theoremsq3 8983 The square of 3 is 9. (Contributed by NM, 26-Apr-2006.)
(3↑2) = 9

Theoremcu2 8984 The cube of 2 is 8. (Contributed by NM, 2-Aug-2004.)
(2↑3) = 8

Theoremirec 8985 The reciprocal of i. (Contributed by NM, 11-Oct-1999.)
(1 / i) = -i

Theoremi2 8986 i squared. (Contributed by NM, 6-May-1999.)
(i↑2) = -1

Theoremi3 8987 i cubed. (Contributed by NM, 31-Jan-2007.)
(i↑3) = -i

Theoremi4 8988 i to the fourth power. (Contributed by NM, 31-Jan-2007.)
(i↑4) = 1

Theoremnnlesq 8989 A positive integer is less than or equal to its square. (Contributed by NM, 15-Sep-1999.) (Revised by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℕ → 𝑁 ≤ (𝑁↑2))

Theoremexpnass 8990 A counterexample showing that exponentiation is not associative. (Contributed by Stefan Allan and Gérard Lang, 21-Sep-2010.)
((3↑3)↑3) < (3↑(3↑3))

Theoremsubsq 8991 Factor the difference of two squares. (Contributed by NM, 21-Feb-2008.)
((A B ℂ) → ((A↑2) − (B↑2)) = ((A + B) · (AB)))

Theoremsubsq2 8992 Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.)
((A B ℂ) → ((A↑2) − (B↑2)) = (((AB)↑2) + ((2 · B) · (AB))))

Theorembinom2i 8993 The square of a binomial. (Contributed by NM, 11-Aug-1999.)
A     &   B        ((A + B)↑2) = (((A↑2) + (2 · (A · B))) + (B↑2))

Theoremsubsqi 8994 Factor the difference of two squares. (Contributed by NM, 7-Feb-2005.)
A     &   B        ((A↑2) − (B↑2)) = ((A + B) · (AB))

Theorembinom2 8995 The square of a binomial. (Contributed by FL, 10-Dec-2006.)
((A B ℂ) → ((A + B)↑2) = (((A↑2) + (2 · (A · B))) + (B↑2)))

Theorembinom21 8996 Special case of binom2 8995 where B = 1. (Contributed by Scott Fenton, 11-May-2014.)
(A ℂ → ((A + 1)↑2) = (((A↑2) + (2 · A)) + 1))

Theorembinom2sub 8997 Expand the square of a subtraction. (Contributed by Scott Fenton, 10-Jun-2013.)
((A B ℂ) → ((AB)↑2) = (((A↑2) − (2 · (A · B))) + (B↑2)))

Theorembinom2subi 8998 Expand the square of a subtraction. (Contributed by Scott Fenton, 13-Jun-2013.)
A     &   B        ((AB)↑2) = (((A↑2) − (2 · (A · B))) + (B↑2))

Theorembinom3 8999 The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)
((A B ℂ) → ((A + B)↑3) = (((A↑3) + (3 · ((A↑2) · B))) + ((3 · (A · (B↑2))) + (B↑3))))

Theoremzesq 9000 An integer is even iff its square is even. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑁 ℤ → ((𝑁 / 2) ℤ ↔ ((𝑁↑2) / 2) ℤ))

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