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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltaddsubi 7101 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A + B) < 𝐶A < (𝐶B))
 
Theoremlt2addi 7102 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶     &   𝐷        ((A < 𝐶 B < 𝐷) → (A + B) < (𝐶 + 𝐷))
 
Theoremle2addi 7103 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
A     &   B     &   𝐶     &   𝐷        ((A𝐶 B𝐷) → (A + B) ≤ (𝐶 + 𝐷))
 
Theoremgt0ne0d 7104 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φ → 0 < A)       (φA ≠ 0)
 
Theoremlt0ne0d 7105 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(φA < 0)       (φA ≠ 0)
 
Theoremleidd 7106 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φAA)
 
Theoremlt0neg1d 7107 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A < 0 ↔ 0 < -A))
 
Theoremlt0neg2d 7108 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (0 < A ↔ -A < 0))
 
Theoremle0neg1d 7109 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (A ≤ 0 ↔ 0 ≤ -A))
 
Theoremle0neg2d 7110 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → (0 ≤ A ↔ -A ≤ 0))
 
Theoremaddgegt0d 7111 Addition of nonnegative and 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))
 
Theoremaddgt0d 7112 Addition of 2 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))
 
Theoremaddge0d 7113 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → 0 ≤ A)    &   (φ → 0 ≤ B)       (φ → 0 ≤ (A + B))
 
Theoremltnegd 7114 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A < B ↔ -B < -A))
 
Theoremlenegd 7115 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (AB ↔ -B ≤ -A))
 
Theoremltnegcon1d 7116 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → -A < B)       (φ → -B < A)
 
Theoremltnegcon2d 7117 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA < -B)       (φB < -A)
 
Theoremlenegcon1d 7118 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ → -AB)       (φ → -BA)
 
Theoremlenegcon2d 7119 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φA ≤ -B)       (φB ≤ -A)
 
Theoremltaddposd 7120 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < AB < (B + A)))
 
Theoremltaddpos2d 7121 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < AB < (A + B)))
 
Theoremltsubposd 7122 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 < A ↔ (BA) < B))
 
Theoremposdifd 7123 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (A < B ↔ 0 < (BA)))
 
Theoremaddge01d 7124 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ BA ≤ (A + B)))
 
Theoremaddge02d 7125 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ BA ≤ (B + A)))
 
Theoremsubge0d 7126 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ (AB) ↔ BA))
 
Theoremsuble0d 7127 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → ((AB) ≤ 0 ↔ AB))
 
Theoremsubge02d 7128 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (0 ≤ B ↔ (AB) ≤ A))
 
Theoremltadd1d 7129 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A + 𝐶) < (B + 𝐶)))
 
Theoremleadd1d 7130 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A + 𝐶) ≤ (B + 𝐶)))
 
Theoremleadd2d 7131 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶 + A) ≤ (𝐶 + B)))
 
Theoremltsubaddd 7132 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (𝐶 + B)))
 
Theoremlesubaddd 7133 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (𝐶 + B)))
 
Theoremltsubadd2d 7134 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) < 𝐶A < (B + 𝐶)))
 
Theoremlesubadd2d 7135 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((AB) ≤ 𝐶A ≤ (B + 𝐶)))
 
Theoremltaddsubd 7136 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶A < (𝐶B)))
 
Theoremltaddsub2d 7137 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) < 𝐶B < (𝐶A)))
 
Theoremleaddsub2d 7138 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → ((A + B) ≤ 𝐶B ≤ (𝐶A)))
 
Theoremsubled 7139 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) ≤ 𝐶)       (φ → (A𝐶) ≤ B)
 
Theoremlesubd 7140 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA ≤ (B𝐶))       (φ𝐶 ≤ (BA))
 
Theoremltsub23d 7141 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ → (AB) < 𝐶)       (φ → (A𝐶) < B)
 
Theoremltsub13d 7142 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < (B𝐶))       (φ𝐶 < (BA))
 
Theoremlesub1d 7143 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (A𝐶) ≤ (B𝐶)))
 
Theoremlesub2d 7144 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (AB ↔ (𝐶B) ≤ (𝐶A)))
 
Theoremltsub1d 7145 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (A𝐶) < (B𝐶)))
 
Theoremltsub2d 7146 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (𝐶B) < (𝐶A)))
 
Theoremltadd1dd 7147 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (A + 𝐶) < (B + 𝐶))
 
Theoremltsub1dd 7148 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (A𝐶) < (B𝐶))
 
Theoremltsub2dd 7149 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (𝐶B) < (𝐶A))
 
Theoremleadd1dd 7150 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (A + 𝐶) ≤ (B + 𝐶))
 
Theoremleadd2dd 7151 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (𝐶 + A) ≤ (𝐶 + B))
 
Theoremlesub1dd 7152 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (A𝐶) ≤ (B𝐶))
 
Theoremlesub2dd 7153 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φAB)       (φ → (𝐶B) ≤ (𝐶A))
 
Theoremle2addd 7154 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB𝐷)       (φ → (A + B) ≤ (𝐶 + 𝐷))
 
Theoremle2subd 7155 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB𝐷)       (φ → (A𝐷) ≤ (𝐶B))
 
Theoremltleaddd 7156 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB𝐷)       (φ → (A + B) < (𝐶 + 𝐷))
 
Theoremleltaddd 7157 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA𝐶)    &   (φB < 𝐷)       (φ → (A + B) < (𝐶 + 𝐷))
 
Theoremlt2addd 7158 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB < 𝐷)       (φ → (A + B) < (𝐶 + 𝐷))
 
Theoremlt2subd 7159 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φ𝐷 ℝ)    &   (φA < 𝐶)    &   (φB < 𝐷)       (φ → (A𝐷) < (𝐶B))
 
Theoremltaddsublt 7160 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
((A B 𝐶 ℝ) → (B < 𝐶 ↔ ((A + B) − 𝐶) < A))
 
Theorem1le1 7161 1 ≤ 1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
1 ≤ 1
 
Theoremgt0add 7162 A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
((A B 0 < (A + B)) → (0 < A 0 < B))
 
3.3.5  Real Apartness
 
Syntaxcreap 7163 Class of real apartness relation.
class #
 
Definitiondf-reap 7164* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨x, y⟩ ∣ ((x y ℝ) (x < y y < x))}
 
Theoremreapval 7165 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7177 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
((A B ℝ) → (A # B ↔ (A < B B < A)))
 
Theoremreapirr 7166 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT]], p. (varies). Beyond the development of # itself, proofs should use apirr 7189 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
(A ℝ → ¬ A # A)
 
Theoremrecexre 7167* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
((A A # 0) → x ℝ (A · x) = 1)
 
Theoremreapti 7168 Real apartness is tight. (Contributed by Jim Kingdon, 30-Jan-2020.)
((A B ℝ) → (A = B ↔ ¬ A # B))
 
3.3.6  Reciprocals
 
Theoremixi 7169 i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(i · i) = -1
 
Theoremrecexgt0 7170* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
((A 0 < A) → x ℝ (0 < x (A · x) = 1))
 
3.3.7  Complex Apartness
 
Syntaxcap 7171 Class of complex apartness relation.
class #
 
Definitiondf-ap 7172* Define complex apartness. Definition 6.1 of Skeleton for the Proof development leading to the. Fundamental Theorem of Algebra, Herman Geuvers, Randy Pollack, Freek Wiedijk, Jan Zwanenburg, October 2, 2000. (Contributed by Jim Kingdon, 26-Jan-2020.)
# = {⟨x, y⟩ ∣ 𝑟 𝑠 𝑡 u ℝ ((x = (𝑟 + (i · 𝑠)) y = (𝑡 + (i · u))) (𝑟 # 𝑡 𝑠 # u))}
 
Theoreminelr 7173 The imaginary unit i is not a real number. (Contributed by NM, 6-May-1999.)
¬ i
 
Theoremrimul 7174 A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((A (i · A) ℝ) → A = 0)
 
Theoremrereim 7175 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
(((A B ℝ) (𝐶 A = (B + (i · 𝐶)))) → (B = A 𝐶 = 0))
 
Theoremapreap 7176 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
((A B ℝ) → (A # BA # B))
 
Theoremreaplt 7177 Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT]], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.)
((A B ℝ) → (A # B ↔ (A < B B < A)))
 
Theoremltmul1a 7178 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
(((A B (𝐶 0 < 𝐶)) A < B) → (A · 𝐶) < (B · 𝐶))
 
Theoremltmul1 7179 Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT]], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B (𝐶 0 < 𝐶)) → (A < B ↔ (A · 𝐶) < (B · 𝐶)))
 
Theoremlemul1 7180 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
((A B (𝐶 0 < 𝐶)) → (AB ↔ (A · 𝐶) ≤ (B · 𝐶)))
 
Theoremreapmul1lem 7181 Lemma for reapmul1 7182. (Contributed by Jim Kingdon, 8-Feb-2020.)
((A B (𝐶 0 < 𝐶)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremreapmul1 7182 Multiplication of both sides of real apartness by a real number apart from zero. (Contributed by Jim Kingdon, 8-Feb-2020.)
((A B (𝐶 𝐶 # 0)) → (A # B ↔ (A · 𝐶) # (B · 𝐶)))
 
Theoremreapadd1 7183 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B 𝐶 ℝ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))
 
Theoremreapneg 7184 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B ℝ) → (A # B ↔ -A # -B))
 
Theoremreapcotr 7185 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℝ) → (A # B → (A # 𝐶 B # 𝐶)))
 
Theoremapsqgt0 7186 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
((A A # 0) → 0 < (A · A))
 
Theoremcru 7187 The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A + (i · B)) = (𝐶 + (i · 𝐷)) ↔ (A = 𝐶 B = 𝐷)))
 
Theoremapreim 7188 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A + (i · B)) # (𝐶 + (i · 𝐷)) ↔ (A # 𝐶 B # 𝐷)))
 
Theoremapirr 7189 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
(A ℂ → ¬ A # A)
 
Theoremapsym 7190 Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B ℂ) → (A # BB # A))
 
Theoremapcotr 7191 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℂ) → (A # B → (A # 𝐶 B # 𝐶)))
 
Theoremapadd1 7192 Addition respects apartness. Analogue of addcan 6793 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
((A B 𝐶 ℂ) → (A # B ↔ (A + 𝐶) # (B + 𝐶)))
 
Theoremapadd2 7193 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
((A B 𝐶 ℂ) → (A # B ↔ (𝐶 + A) # (𝐶 + B)))
 
Theoremextadd 7194 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5441. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) # (𝐶 + 𝐷) → (A # 𝐶 B # 𝐷)))
 
Theoremapneg 7195 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
((A B ℂ) → (A # B ↔ -A # -B))
 
PART 4  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
 
4.1  Mathboxes for user contributions
 
4.1.1  Mathbox guidelines
 
Theoremmathbox 7196 (This theorem is a dummy placeholder for these guidelines. The name of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of set.mm.

Mathboxes are provided to help keep your work synchronized with changes in set.mm, but they shouldn't be depended on as a permanent archive. If you want to preserve your original contribution, it is your responsibility to keep your own copy of it along with the version of set.mm that works with it.

Guidelines:

1. If at all possible, please use only 0-ary class constants for new definitions.

2. Try to follow the style of the rest of set.mm. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The metamath program command "write source set.mm /rewrap" will take care of wrapping comment lines and indentation conventions. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

3. Before submitting a revised mathbox, please make sure it verifies against the current set.mm.

4. Mathboxes should be independent i.e. the proofs should verify with all other mathboxes removed. If you need a theorem from another mathbox, that is fine (and encouraged), but let me know, so I can move the theorem to the main section. One way avoid undesired accidental use of other mathbox theorems is to develop your mathbox using a modified set.mm that has mathboxes removed.

Notes:

1. We may decide to move some theorems to the main part of set.mm for general use.

2. We may make changes to mathboxes to maintain the overall quality of set.mm. Normally we will let you know if a change might impact what you are working on.

3. If you use theorems from another user's mathbox, we don't provide assurance that they are based on correct or consistent $a statements. (If you find such a problem, please let us know so it can be corrected.) (Contributed by NM, 20-Feb-2007.) (New usage is discouraged.)

x = x
 
4.2  Mathbox for Mykola Mostovenko
 
Theoremax1hfs 7197 Heyting's formal system Axiom #1 from [Heyting] p. 127. (Contributed by MM, 11-Aug-2018.)
(φ → (φ φ))
 
4.3  Mathbox for BJ
 
4.3.1  Propositional calculus
 
Theoremnnexmid 7198 Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but, of course, does not prove excluded middle) for any formula. (Contributed by BJ, 9-Oct-2019.)
¬ ¬ (φ ¬ φ)
 
Theoremnndc 7199 Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but, of course, does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)
¬ ¬ DECID φ
 
Theoremdcdc 7200 Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)
(DECID DECID φDECID φ)
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