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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddsub12d 7101 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A + (B𝐶)) = (B + (A𝐶)))
 
Theoremnpncand 7102 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (B𝐶)) = (A𝐶))
 
Theoremnppcand 7103 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (((AB) + 𝐶) + B) = (A + 𝐶))
 
Theoremnppcan2d 7104 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A − (B + 𝐶)) + 𝐶) = (AB))
 
Theoremnppcan3d 7105 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (𝐶 + B)) = (A + 𝐶))
 
Theoremsubsubd 7106 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = ((AB) + 𝐶))
 
Theoremsubsub2d 7107 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = (A + (𝐶B)))
 
Theoremsubsub3d 7108 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A − (B𝐶)) = ((A + 𝐶) − B))
 
Theoremsubsub4d 7109 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − 𝐶) = (A − (B + 𝐶)))
 
Theoremsub32d 7110 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − 𝐶) = ((A𝐶) − B))
 
Theoremnnncand 7111 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A − (B𝐶)) − 𝐶) = (AB))
 
Theoremnnncan1d 7112 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) − (A𝐶)) = (𝐶B))
 
Theoremnnncan2d 7113 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A𝐶) − (B𝐶)) = (AB))
 
Theoremnpncan3d 7114 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) + (𝐶A)) = (𝐶B))
 
Theorempnpcand 7115 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − (A + 𝐶)) = (B𝐶))
 
Theorempnpcan2d 7116 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + 𝐶) − (B + 𝐶)) = (AB))
 
Theorempnncand 7117 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) − (A𝐶)) = (B + 𝐶))
 
Theoremppncand 7118 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A + B) + (𝐶B)) = (A + 𝐶))
 
Theoremsubcand 7119 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (AB) = (A𝐶))       (φB = 𝐶)
 
Theoremsubcan2d 7120 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ → (A𝐶) = (B𝐶))       (φA = B)
 
Theoremsubcanad 7121 Cancellation law for subtraction. Deduction form of subcan 7022. Generalization of subcand 7119. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) = (A𝐶) ↔ B = 𝐶))
 
Theoremsubneintrd 7122 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7119. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φB𝐶)       (φ → (AB) ≠ (A𝐶))
 
Theoremsubcan2ad 7123 Cancellation law for subtraction. Deduction form of subcan2 6992. Generalization of subcan2d 7120. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((A𝐶) = (B𝐶) ↔ A = B))
 
Theoremsubneintr2d 7124 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7120. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φAB)       (φ → (A𝐶) ≠ (B𝐶))
 
Theoremaddsub4d 7125 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) − (𝐶 + 𝐷)) = ((A𝐶) + (B𝐷)))
 
Theoremsubadd4d 7126 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((AB) − (𝐶𝐷)) = ((A + 𝐷) − (B + 𝐶)))
 
Theoremsub4d 7127 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((AB) − (𝐶𝐷)) = ((A𝐶) − (B𝐷)))
 
Theorem2addsubd 7128 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → (((A + B) + 𝐶) − 𝐷) = (((A + 𝐶) − 𝐷) + B))
 
Theoremaddsubeq4d 7129 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) = (𝐶 + 𝐷) ↔ (𝐶A) = (B𝐷)))
 
Theoremsubeqrev 7130 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) = (𝐶𝐷) ↔ (BA) = (𝐷𝐶)))
 
Theorempncan1 7131 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(A ℂ → ((A + 1) − 1) = A)
 
Theoremnpcan1 7132 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
(A ℂ → ((A − 1) + 1) = A)
 
Theoremsubeq0bd 7133 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7088. Converse of subeq0d 7086. Contrapositive of subne0ad 7089. (Contributed by David Moews, 28-Feb-2017.)
(φA ℂ)    &   (φA = B)       (φ → (AB) = 0)
 
Theoremrenegcld 7134 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)       (φ → -A ℝ)
 
Theoremresubcld 7135 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)       (φ → (AB) ℝ)
 
3.3.3  Multiplication
 
Theoremkcnktkm1cn 7136 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
(𝐾 ℂ → (𝐾 · (𝐾 − 1)) ℂ)
 
Theoremmuladd 7137 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) · (𝐶 + 𝐷)) = (((A · 𝐶) + (𝐷 · B)) + ((A · 𝐷) + (𝐶 · B))))
 
Theoremsubdi 7138 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
((A B 𝐶 ℂ) → (A · (B𝐶)) = ((A · B) − (A · 𝐶)))
 
Theoremsubdir 7139 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
((A B 𝐶 ℂ) → ((AB) · 𝐶) = ((A · 𝐶) − (B · 𝐶)))
 
Theoremmul02 7140 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
(A ℂ → (0 · A) = 0)
 
Theoremmul02lem2 7141 Zero times a real is zero. Although we prove it as a corollary of mul02 7140, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7140. (Contributed by Scott Fenton, 3-Jan-2013.)
(A ℝ → (0 · A) = 0)
 
Theoremmul01 7142 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
(A ℂ → (A · 0) = 0)
 
Theoremmul02i 7143 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
A        (0 · A) = 0
 
Theoremmul01i 7144 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
A        (A · 0) = 0
 
Theoremmul02d 7145 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (0 · A) = 0)
 
Theoremmul01d 7146 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (A · 0) = 0)
 
Theoremine0 7147 The imaginary unit i is not zero. (Contributed by NM, 6-May-1999.)
i ≠ 0
 
Theoremmulneg1 7148 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℂ) → (-A · B) = -(A · B))
 
Theoremmulneg2 7149 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
((A B ℂ) → (A · -B) = -(A · B))
 
Theoremmulneg12 7150 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
((A B ℂ) → (-A · B) = (A · -B))
 
Theoremmul2neg 7151 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℂ) → (-A · -B) = (A · B))
 
Theoremsubmul2 7152 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
((A B 𝐶 ℂ) → (A − (B · 𝐶)) = (A + (B · -𝐶)))
 
Theoremmulm1 7153 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
(A ℂ → (-1 · A) = -A)
 
Theoremmulsub 7154 Product of two differences. (Contributed by NM, 14-Jan-2006.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) · (𝐶𝐷)) = (((A · 𝐶) + (𝐷 · B)) − ((A · 𝐷) + (𝐶 · B))))
 
Theoremmulsub2 7155 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
(((A B ℂ) (𝐶 𝐷 ℂ)) → ((AB) · (𝐶𝐷)) = ((BA) · (𝐷𝐶)))
 
Theoremmulm1i 7156 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
A        (-1 · A) = -A
 
Theoremmulneg1i 7157 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (-A · B) = -(A · B)
 
Theoremmulneg2i 7158 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (A · -B) = -(A · B)
 
Theoremmul2negi 7159 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        (-A · -B) = (A · B)
 
Theoremsubdii 7160 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
A     &   B     &   𝐶        (A · (B𝐶)) = ((A · B) − (A · 𝐶))
 
Theoremsubdiri 7161 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
A     &   B     &   𝐶        ((AB) · 𝐶) = ((A · 𝐶) − (B · 𝐶))
 
Theoremmuladdi 7162 Product of two sums. (Contributed by NM, 17-May-1999.)
A     &   B     &   𝐶     &   𝐷        ((A + B) · (𝐶 + 𝐷)) = (((A · 𝐶) + (𝐷 · B)) + ((A · 𝐷) + (𝐶 · B)))
 
Theoremmulm1d 7163 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (-1 · A) = -A)
 
Theoremmulneg1d 7164 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A · B) = -(A · B))
 
Theoremmulneg2d 7165 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · -B) = -(A · B))
 
Theoremmul2negd 7166 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (-A · -B) = (A · B))
 
Theoremsubdid 7167 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → (A · (B𝐶)) = ((A · B) − (A · 𝐶)))
 
Theoremsubdird 7168 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)       (φ → ((AB) · 𝐶) = ((A · 𝐶) − (B · 𝐶)))
 
Theoremmuladdd 7169 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((A + B) · (𝐶 + 𝐷)) = (((A · 𝐶) + (𝐷 · B)) + ((A · 𝐷) + (𝐶 · B))))
 
Theoremmulsubd 7170 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)    &   (φ𝐶 ℂ)    &   (φ𝐷 ℂ)       (φ → ((AB) · (𝐶𝐷)) = (((A · 𝐶) + (𝐷 · B)) − ((A · 𝐷) + (𝐶 · B))))
 
Theoremmulsubfacd 7171 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
(φA ℂ)    &   (φB ℂ)       (φ → ((A · B) − B) = ((A − 1) · B))
 
3.3.4  Ordering on reals (cont.)
 
Theoremltadd2 7172 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (𝐶 + A) < (𝐶 + B)))
 
Theoremltadd2i 7173 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A < B ↔ (𝐶 + A) < (𝐶 + B))
 
Theoremltadd2d 7174 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)       (φ → (A < B ↔ (𝐶 + A) < (𝐶 + B)))
 
Theoremltadd2dd 7175 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)       (φ → (𝐶 + A) < (𝐶 + B))
 
Theoremltletrd 7176 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
(φA ℝ)    &   (φB ℝ)    &   (φ𝐶 ℝ)    &   (φA < B)    &   (φB𝐶)       (φA < 𝐶)
 
Theoremgt0ne0 7177 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A 0 < A) → A ≠ 0)
 
Theoremlt0ne0 7178 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((A A < 0) → A ≠ 0)
 
Theoremltadd1 7179 Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (A + 𝐶) < (B + 𝐶)))
 
Theoremleadd1 7180 Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (AB ↔ (A + 𝐶) ≤ (B + 𝐶)))
 
Theoremleadd2 7181 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
((A B 𝐶 ℝ) → (AB ↔ (𝐶 + A) ≤ (𝐶 + B)))
 
Theoremltsubadd 7182 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → ((AB) < 𝐶A < (𝐶 + B)))
 
Theoremltsubadd2 7183 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((A B 𝐶 ℝ) → ((AB) < 𝐶A < (B + 𝐶)))
 
Theoremlesubadd 7184 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → ((AB) ≤ 𝐶A ≤ (𝐶 + B)))
 
Theoremlesubadd2 7185 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
((A B 𝐶 ℝ) → ((AB) ≤ 𝐶A ≤ (B + 𝐶)))
 
Theoremltaddsub 7186 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → ((A + B) < 𝐶A < (𝐶B)))
 
Theoremltaddsub2 7187 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → ((A + B) < 𝐶B < (𝐶A)))
 
Theoremleaddsub 7188 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((A B 𝐶 ℝ) → ((A + B) ≤ 𝐶A ≤ (𝐶B)))
 
Theoremleaddsub2 7189 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
((A B 𝐶 ℝ) → ((A + B) ≤ 𝐶B ≤ (𝐶A)))
 
Theoremsuble 7190 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((A B 𝐶 ℝ) → ((AB) ≤ 𝐶 ↔ (A𝐶) ≤ B))
 
Theoremlesub 7191 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B 𝐶 ℝ) → (A ≤ (B𝐶) ↔ 𝐶 ≤ (BA)))
 
Theoremltsub23 7192 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((A B 𝐶 ℝ) → ((AB) < 𝐶 ↔ (A𝐶) < B))
 
Theoremltsub13 7193 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → (A < (B𝐶) ↔ 𝐶 < (BA)))
 
Theoremle2add 7194 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A𝐶 B𝐷) → (A + B) ≤ (𝐶 + 𝐷)))
 
Theoremlt2add 7195 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A < 𝐶 B < 𝐷) → (A + B) < (𝐶 + 𝐷)))
 
Theoremltleadd 7196 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A < 𝐶 B𝐷) → (A + B) < (𝐶 + 𝐷)))
 
Theoremleltadd 7197 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A𝐶 B < 𝐷) → (A + B) < (𝐶 + 𝐷)))
 
Theoremaddgt0 7198 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℝ) (0 < A 0 < B)) → 0 < (A + B))
 
Theoremaddgegt0 7199 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℝ) (0 ≤ A 0 < B)) → 0 < (A + B))
 
Theoremaddgtge0 7200 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℝ) (0 < A 0 ≤ B)) → 0 < (A + B))
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