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Theorem List for Intuitionistic Logic Explorer - 7201-7300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsubdiri 7201 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 7202 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 7203 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)       (φ → (-1 · A) = -A)

Theoremmulneg1d 7204 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 7205 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
(φA ℂ)    &   (φB ℂ)       (φ → (A · -B) = -(A · B))

Theoremmul2negd 7206 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 7207 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 7208 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 7209 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 7210 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 7211 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 7212 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 7213 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A < B ↔ (𝐶 + A) < (𝐶 + B))

Theoremltadd2d 7214 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 7215 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 7216 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 7217 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A 0 < A) → A ≠ 0)

Theoremlt0ne0 7218 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
((A A < 0) → A ≠ 0)

Theoremltadd1 7219 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 7220 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 7221 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 7222 '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 7223 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
((A B 𝐶 ℝ) → ((AB) < 𝐶A < (B + 𝐶)))

Theoremlesubadd 7224 '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 7225 '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 7226 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → ((A + B) < 𝐶A < (𝐶B)))

Theoremltaddsub2 7227 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → ((A + B) < 𝐶B < (𝐶A)))

Theoremleaddsub 7228 '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 7229 '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 7230 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
((A B 𝐶 ℝ) → ((AB) ≤ 𝐶 ↔ (A𝐶) ≤ B))

Theoremlesub 7231 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 7232 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
((A B 𝐶 ℝ) → ((AB) < 𝐶 ↔ (A𝐶) < B))

Theoremltsub13 7233 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
((A B 𝐶 ℝ) → (A < (B𝐶) ↔ 𝐶 < (BA)))

Theoremle2add 7234 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 7235 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 7236 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 7237 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 7238 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 7239 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 7240 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))

Theoremaddge0 7241 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
(((A B ℝ) (0 ≤ A 0 ≤ B)) → 0 ≤ (A + B))

Theoremltaddpos 7242 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
((A B ℝ) → (0 < AB < (B + A)))

Theoremltaddpos2 7243 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
((A B ℝ) → (0 < AB < (A + B)))

Theoremltsubpos 7244 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
((A B ℝ) → (0 < A ↔ (BA) < B))

Theoremposdif 7245 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
((A B ℝ) → (A < B ↔ 0 < (BA)))

Theoremlesub1 7246 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (AB ↔ (A𝐶) ≤ (B𝐶)))

Theoremlesub2 7247 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (AB ↔ (𝐶B) ≤ (𝐶A)))

Theoremltsub1 7248 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (A𝐶) < (B𝐶)))

Theoremltsub2 7249 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B 𝐶 ℝ) → (A < B ↔ (𝐶B) < (𝐶A)))

Theoremlt2sub 7250 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A < 𝐶 𝐷 < B) → (AB) < (𝐶𝐷)))

Theoremle2sub 7251 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
(((A B ℝ) (𝐶 𝐷 ℝ)) → ((A𝐶 𝐷B) → (AB) ≤ (𝐶𝐷)))

Theoremltneg 7252 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (A < B ↔ -B < -A))

Theoremltnegcon1 7253 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
((A B ℝ) → (-A < B ↔ -B < A))

Theoremltnegcon2 7254 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
((A B ℝ) → (A < -BB < -A))

Theoremleneg 7255 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (AB ↔ -B ≤ -A))

Theoremlenegcon1 7256 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
((A B ℝ) → (-AB ↔ -BA))

Theoremlenegcon2 7257 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
((A B ℝ) → (A ≤ -BB ≤ -A))

Theoremlt0neg1 7258 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
(A ℝ → (A < 0 ↔ 0 < -A))

Theoremlt0neg2 7259 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(A ℝ → (0 < A ↔ -A < 0))

Theoremle0neg1 7260 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
(A ℝ → (A ≤ 0 ↔ 0 ≤ -A))

Theoremle0neg2 7261 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
(A ℝ → (0 ≤ A ↔ -A ≤ 0))

Theoremaddge01 7262 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
((A B ℝ) → (0 ≤ BA ≤ (A + B)))

Theoremaddge02 7263 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
((A B ℝ) → (0 ≤ BA ≤ (B + A)))

Theoremadd20 7264 Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)
(((A 0 ≤ A) (B 0 ≤ B)) → ((A + B) = 0 ↔ (A = 0 B = 0)))

Theoremsubge0 7265 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → (0 ≤ (AB) ↔ BA))

Theoremsuble0 7266 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → ((AB) ≤ 0 ↔ AB))

Theoremleaddle0 7267 The sum of a real number and a second real number is less then the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
((A B ℝ) → ((A + B) ≤ AB ≤ 0))

Theoremsubge02 7268 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
((A B ℝ) → (0 ≤ B ↔ (AB) ≤ A))

Theoremlesub0 7269 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
((A B ℝ) → ((0 ≤ A B ≤ (BA)) ↔ A = 0))

Theoremmullt0 7270 The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
(((A A < 0) (B B < 0)) → 0 < (A · B))

Theorem0le1 7271 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
0 ≤ 1

Theoremleidi 7272 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
A        AA

Theoremgt0ne0i 7273 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
A        (0 < AA ≠ 0)

Theoremgt0ne0ii 7274 Positive implies nonzero. (Contributed by NM, 15-May-1999.)
A     &   0 < A       A ≠ 0

Theoremaddgt0i 7275 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 < A 0 < B) → 0 < (A + B))

Theoremaddge0i 7276 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 ≤ A 0 ≤ B) → 0 ≤ (A + B))

Theoremaddgegt0i 7277 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
A     &   B        ((0 ≤ A 0 < B) → 0 < (A + B))

Theoremaddgt0ii 7278 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
A     &   B     &   0 < A    &   0 < B       0 < (A + B)

Theoremadd20i 7279 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
A     &   B        ((0 ≤ A 0 ≤ B) → ((A + B) = 0 ↔ (A = 0 B = 0)))

Theoremltnegi 7280 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
A     &   B        (A < B ↔ -B < -A)

Theoremlenegi 7281 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
A     &   B        (AB ↔ -B ≤ -A)

Theoremltnegcon2i 7282 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
A     &   B        (A < -BB < -A)

Theoremlesub0i 7283 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B        ((0 ≤ A B ≤ (BA)) ↔ A = 0)

Theoremltaddposi 7284 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
A     &   B        (0 < AB < (B + A))

Theoremposdifi 7285 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
A     &   B        (A < B ↔ 0 < (BA))

Theoremltnegcon1i 7286 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
A     &   B        (-A < B ↔ -B < A)

Theoremlenegcon1i 7287 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
A     &   B        (-AB ↔ -BA)

Theoremsubge0i 7288 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
A     &   B        (0 ≤ (AB) ↔ BA)

Theoremltadd1i 7289 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        (A < B ↔ (A + 𝐶) < (B + 𝐶))

Theoremleadd1i 7290 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
A     &   B     &   𝐶        (AB ↔ (A + 𝐶) ≤ (B + 𝐶))

Theoremleadd2i 7291 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
A     &   B     &   𝐶        (AB ↔ (𝐶 + A) ≤ (𝐶 + B))

Theoremltsubaddi 7292 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B     &   𝐶        ((AB) < 𝐶A < (𝐶 + B))

Theoremlesubaddi 7293 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
A     &   B     &   𝐶        ((AB) ≤ 𝐶A ≤ (𝐶 + B))

Theoremltsubadd2i 7294 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
A     &   B     &   𝐶        ((AB) < 𝐶A < (B + 𝐶))

Theoremlesubadd2i 7295 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
A     &   B     &   𝐶        ((AB) ≤ 𝐶A ≤ (B + 𝐶))

Theoremltaddsubi 7296 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
A     &   B     &   𝐶        ((A + B) < 𝐶A < (𝐶B))

Theoremlt2addi 7297 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 7298 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)
A     &   B     &   𝐶     &   𝐷        ((A𝐶 B𝐷) → (A + B) ≤ (𝐶 + 𝐷))

Theoremgt0ne0d 7299 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
(φ → 0 < A)       (φA ≠ 0)

Theoremlt0ne0d 7300 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)
(φA < 0)       (φA ≠ 0)

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