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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltnsym 6901 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)
((A B ℝ) → (A < B → ¬ B < A))

Theoremltle 6902 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)
((A B ℝ) → (A < BAB))

Theoremlelttr 6903 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 23-May-1999.)
((A B 𝐶 ℝ) → ((AB B < 𝐶) → A < 𝐶))

Theoremltletr 6904 Transitive law. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 25-Aug-1999.)
((A B 𝐶 ℝ) → ((A < B B𝐶) → A < 𝐶))

Theoremltnsym2 6905 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℝ) → ¬ (A < B B < A))

Theoremeqle 6906 Equality implies 'less than or equal to'. (Contributed by NM, 4-Apr-2005.)
((A A = B) → AB)

Theoremltnri 6907 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)
A         ¬ A < A

Theoremeqlei 6908 Equality implies 'less than or equal to'. (Contributed by NM, 23-May-1999.) (Revised by Alexander van der Vekens, 20-Mar-2018.)
A        (A = BAB)

Theoremeqlei2 6909 Equality implies 'less than or equal to'. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
A        (B = ABA)

Theoremgtneii 6910 'Less than' implies not equal. (Contributed by Mario Carneiro, 30-Sep-2013.)
A     &   A < B       BA

Theoremltneii 6911 'Greater than' implies not equal. (Contributed by Mario Carneiro, 16-Sep-2015.)
A     &   A < B       AB

Theoremlttri3i 6912 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
A     &   B        (A = B ↔ (¬ A < B ¬ B < A))

Theoremletri3i 6913 Tightness of real apartness. (Contributed by NM, 14-May-1999.)
A     &   B        (A = B ↔ (AB BA))

Theoremltnsymi 6914 'Less than' is not symmetric. (Contributed by NM, 6-May-1999.)
A     &   B        (A < B → ¬ B < A)

Theoremlenlti 6915 'Less than or equal to' in terms of 'less than'. (Contributed by NM, 24-May-1999.)
A     &   B        (AB ↔ ¬ B < A)

Theoremltlei 6916 'Less than' implies 'less than or equal to'. (Contributed by NM, 14-May-1999.)
A     &   B        (A < BAB)

Theoremltleii 6917 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.)
A     &   B     &   A < B       AB

Theoremltnei 6918 'Less than' implies not equal. (Contributed by NM, 28-Jul-1999.)
A     &   B        (A < BBA)

Theoremlttri 6919 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A < B B < 𝐶) → A < 𝐶)

Theoremlelttri 6920 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((AB B < 𝐶) → A < 𝐶)

Theoremltletri 6921 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A < B B𝐶) → A < 𝐶)

Theoremletri 6922 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((AB B𝐶) → A𝐶)

Theoremle2tri3i 6923 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)
A     &   B     &   𝐶        ((AB B𝐶 𝐶A) ↔ (A = B B = 𝐶 𝐶 = A))

Theoremmulgt0i 6924 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
A     &   B        ((0 < A 0 < B) → 0 < (A · B))

Theoremmulgt0ii 6925 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A · B)

Theoremltnrd 6926 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → ¬ A < A)

Theoremgtned 6927 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φBA)

Theoremltned 6928 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φA < B)       (φAB)

Theoremlttri3d 6929 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (¬ A < B ¬ B < A)))

Theoremletri3d 6930 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A = B ↔ (AB BA)))

Theoremlenltd 6931 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (AB ↔ ¬ B < A))

Theoremltled 6932 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φAB)

Theoremltnsymd 6933 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < B)       (φ → ¬ B < A)

Theoremmulgt0d 6934 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 < A)    &   (φ → 0 < B)       (φ → 0 < (A · B))

Theoremletrd 6935 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)    &   (φB𝐶)       (φA𝐶)

Theoremlelttrd 6936 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)    &   (φB < 𝐶)       (φA < 𝐶)

Theoremlttrd 6937 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)    &   (φB < 𝐶)       (φA < 𝐶)

Theorem0lt1 6938 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
0 < 1

3.2.5  Initial properties of the complex numbers

Theoremmul12 6939 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
((A B 𝐶 ℂ) → (A · (B · 𝐶)) = (B · (A · 𝐶)))

Theoremmul32 6940 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = ((A · 𝐶) · B))

Theoremmul31 6941 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℂ) → ((A · B) · 𝐶) = ((𝐶 · B) · A))

Theoremmul4 6942 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A · B) · (𝐶 · 𝐷)) = ((A · 𝐶) · (B · 𝐷)))

Theoremmuladd11 6943 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
((A B ℂ) → ((1 + A) · (1 + B)) = ((1 + A) + (B + (A · B))))

Theorem1p1times 6944 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(A ℂ → ((1 + 1) · A) = (A + A))

Theorempeano2cn 6945 A theorem for complex numbers analogous the second Peano postulate peano2 4261. (Contributed by NM, 17-Aug-2005.)
(A ℂ → (A + 1) ℂ)

Theorempeano2re 6946 A theorem for reals analogous the second Peano postulate peano2 4261. (Contributed by NM, 5-Jul-2005.)
(A ℝ → (A + 1) ℝ)

((A B ℂ) → (A + B) = (B + A))

Theoremaddid1 6948 0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
(A ℂ → (A + 0) = A)

Theoremaddid2 6949 0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℂ → (0 + A) = A)

Theoremreaddcan 6950 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℝ) → ((𝐶 + A) = (𝐶 + B) ↔ A = B))

Theorem00id 6951 0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
(0 + 0) = 0

Theoremaddid1i 6952 0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
A        (A + 0) = A

Theoremaddid2i 6953 0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
A        (0 + A) = A

Theoremaddcomi 6954 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
A     &   B        (A + B) = (B + A)

A     &   B     &   (A + B) = 𝐶       (B + A) = 𝐶

Theoremmul12i 6956 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B     &   𝐶        (A · (B · 𝐶)) = (B · (A · 𝐶))

Theoremmul32i 6957 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
A     &   B     &   𝐶        ((A · B) · 𝐶) = ((A · 𝐶) · B)

Theoremmul4i 6958 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
A     &   B     &   𝐶     &   𝐷        ((A · B) · (𝐶 · 𝐷)) = ((A · 𝐶) · (B · 𝐷))

Theoremaddid1d 6959 0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A + 0) = A)

Theoremaddid2d 6960 0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (0 + A) = A)

Theoremaddcomd 6961 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A + B) = (B + A))

Theoremmul12d 6962 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A · (B · 𝐶)) = (B · (A · 𝐶)))

Theoremmul32d 6963 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A · B) · 𝐶) = ((A · 𝐶) · B))

Theoremmul31d 6964 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A · B) · 𝐶) = ((𝐶 · B) · A))

Theoremmul4d 6965 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A · B) · (𝐶 · 𝐷)) = ((A · 𝐶) · (B · 𝐷)))

3.3  Real and complex numbers - basic operations

Theoremadd12 6966 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
((A B 𝐶 ℂ) → (A + (B + 𝐶)) = (B + (A + 𝐶)))

Theoremadd32 6967 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
((A B 𝐶 ℂ) → ((A + B) + 𝐶) = ((A + 𝐶) + B))

Theoremadd32r 6968 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
((A B 𝐶 ℂ) → (A + (B + 𝐶)) = ((A + 𝐶) + B))

Theoremadd4 6969 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))

Theoremadd42 6970 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (𝐷 + B)))

Theoremadd12i 6971 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A + (B + 𝐶)) = (B + (A + 𝐶))

Theoremadd32i 6972 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        ((A + B) + 𝐶) = ((A + 𝐶) + B)

Theoremadd4i 6973 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷))

Theoremadd42i 6974 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (𝐷 + B))

Theoremadd12d 6975 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A + (B + 𝐶)) = (B + (A + 𝐶)))

Theoremadd32d 6976 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) + 𝐶) = ((A + 𝐶) + B))

Theoremadd4d 6977 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))

Theoremadd42d 6978 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (𝐷 + B)))

3.3.2  Subtraction

Syntaxcmin 6979 Extend class notation to include subtraction.
class

Syntaxcneg 6980 Extend class notation to include unary minus. The symbol - is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus (-) and subtraction cmin 6979 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5455, if we used the same symbol then "( − AB) " could mean either "A " minus "B", or it could represent the (meaningless) operation of classes " " and "B " connected with "operation" "A". On the other hand, "(-AB) " is unambiguous.
class -A

Definitiondf-sub 6981* Define subtraction. Theorem subval 7000 shows its value (and describes how this definition works), theorem subaddi 7094 relates it to addition, and theorems subcli 7083 and resubcli 7070 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
− = (x ℂ, y ℂ ↦ (z ℂ (y + z) = x))

Definitiondf-neg 6982 Define the negative of a number (unary minus). We use different symbols for unary minus (-) and subtraction () to prevent syntax ambiguity. See cneg 6980 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
-A = (0 − A)

Theoremcnegexlem1 6983 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 6986. (Contributed by Eric Schmidt, 22-May-2007.)
((A B 𝐶 ℂ) → ((A + B) = (A + 𝐶) ↔ B = 𝐶))

Theoremcnegexlem2 6984 Existence of a real number which produces a real number when multiplied by i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 6986. (Contributed by Eric Schmidt, 22-May-2007.)
y ℝ (i · y)

Theoremcnegexlem3 6985* Existence of real number difference. Lemma for cnegex 6986. (Contributed by Eric Schmidt, 22-May-2007.)
((𝑏 y ℝ) → 𝑐 ℝ (𝑏 + 𝑐) = y)

Theoremcnegex 6986* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
(A ℂ → x ℂ (A + x) = 0)

Theoremcnegex2 6987* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℂ → x ℂ (x + A) = 0)

Theoremaddcan 6988 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℂ) → ((A + B) = (A + 𝐶) ↔ B = 𝐶))

Theoremaddcan2 6989 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
((A B 𝐶 ℂ) → ((A + 𝐶) = (B + 𝐶) ↔ A = B))

Theoremaddcani 6990 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
A     &   B     &   𝐶        ((A + B) = (A + 𝐶) ↔ B = 𝐶)

Theoremaddcan2i 6991 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
A     &   B     &   𝐶        ((A + 𝐶) = (B + 𝐶) ↔ A = B)

Theoremaddcand 6992 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) = (A + 𝐶) ↔ B = 𝐶))

Theoremaddcan2d 6993 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + 𝐶) = (B + 𝐶) ↔ A = B))

Theoremaddcanad 6994 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 6992. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A + B) = (A + 𝐶))       (φB = 𝐶)

Theoremaddcan2ad 6995 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 6993. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A + 𝐶) = (B + 𝐶))       (φA = B)

Theoremaddneintrd 6996 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 6994. Consequence of addcand 6992. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB𝐶)       (φ → (A + B) ≠ (A + 𝐶))

Theoremaddneintr2d 6997 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 6995. Consequence of addcan2d 6993. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φAB)       (φ → (A + 𝐶) ≠ (B + 𝐶))

Theorem0cnALT 6998 Alternate proof of 0cn 6817. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
0

Theoremnegeu 6999* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℂ) → ∃!x ℂ (A + x) = B)

Theoremsubval 7000* Value of subtraction, which is the (unique) element x such that B + x = A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
((A B ℂ) → (AB) = (x ℂ (B + x) = A))

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