P. 110 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 10901-11000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	addmulsub 10901	The product of a sum and a difference. (Contributed by AV, 5-Mar-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) · (𝐶 − 𝐷)) = (((𝐴 · 𝐶) + (𝐵 · 𝐶)) − ((𝐴 · 𝐷) + (𝐵 · 𝐷))))
Theorem	subaddmulsub 10902	The difference with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ) ∧ 𝐸 ∈ ℂ) → (𝐸 − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐸 − (𝐴 · 𝐶)) − (𝐵 · 𝐶)) + ((𝐴 · 𝐷) + (𝐵 · 𝐷))))
Theorem	mulsubaddmulsub 10903	A special difference of a product with a product of a sum and a difference. (Contributed by AV, 5-Mar-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐵 · 𝐶) − ((𝐴 + 𝐵) · (𝐶 − 𝐷))) = (((𝐴 · 𝐷) + (𝐵 · 𝐷)) − (𝐴 · 𝐶)))
5.3.4  Ordering on reals (cont.)
Theorem	gt0ne0 10904	Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → 𝐴 ≠ 0)
Theorem	lt0ne0 10905	A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)
Theorem	ltadd1 10906	Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶)))
Theorem	leadd1 10907	Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶)))
Theorem	leadd2 10908	Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵)))
Theorem	ltsubadd 10909	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵)))
Theorem	ltsubadd2 10910	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	lesubadd 10911	'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 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵)))
Theorem	lesubadd2 10912	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶)))
Theorem	ltaddsub 10913	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵)))
Theorem	ltaddsub2 10914	'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) < 𝐶 ↔ 𝐵 < (𝐶 − 𝐴)))
Theorem	leaddsub 10915	'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 − 𝐵)))
Theorem	leaddsub2 10916	'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐶 ↔ 𝐵 ≤ (𝐶 − 𝐴)))
Theorem	suble 10917	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 𝐶 ↔ (𝐴 − 𝐶) ≤ 𝐵))
Theorem	lesub 10918	Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ (𝐵 − 𝐶) ↔ 𝐶 ≤ (𝐵 − 𝐴)))
Theorem	ltsub23 10919	'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 − 𝐵) < 𝐶 ↔ (𝐴 − 𝐶) < 𝐵))
Theorem	ltsub13 10920	'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < (𝐵 − 𝐶) ↔ 𝐶 < (𝐵 − 𝐴)))
Theorem	le2add 10921	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 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) ≤ (𝐶 + 𝐷)))
Theorem	ltleadd 10922	Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 ≤ 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
Theorem	leltadd 10923	Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
Theorem	lt2add 10924	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 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐵 < 𝐷) → (𝐴 + 𝐵) < (𝐶 + 𝐷)))
Theorem	addgt0 10925	The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
Theorem	addgegt0 10926	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 < 𝐵)) → 0 < (𝐴 + 𝐵))
Theorem	addgtge0 10927	The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 < 𝐴 ∧ 0 ≤ 𝐵)) → 0 < (𝐴 + 𝐵))
Theorem	addge0 10928	The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 + 𝐵))
Theorem	ltaddpos 10929	Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	ltaddpos2 10930	Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐴 + 𝐵)))
Theorem	ltsubpos 10931	Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ (𝐵 − 𝐴) < 𝐵))
Theorem	posdif 10932	Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
Theorem	lesub1 10933	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 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)))
Theorem	lesub2 10934	Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ (𝐶 − 𝐵) ≤ (𝐶 − 𝐴)))
Theorem	ltsub1 10935	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 − 𝐶) < (𝐵 − 𝐶)))
Theorem	ltsub2 10936	Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐶 − 𝐵) < (𝐶 − 𝐴)))
Theorem	lt2sub 10937	Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 < 𝐶 ∧ 𝐷 < 𝐵) → (𝐴 − 𝐵) < (𝐶 − 𝐷)))
Theorem	le2sub 10938	Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷)))
Theorem	ltneg 10939	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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
Theorem	ltnegcon1 10940	Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴))
Theorem	ltnegcon2 10941	Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < -𝐵 ↔ 𝐵 < -𝐴))
Theorem	leneg 10942	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 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴))
Theorem	lenegcon1 10943	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴))
Theorem	lenegcon2 10944	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴))
Theorem	lt0neg1 10945	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))
Theorem	lt0neg2 10946	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ -𝐴 < 0))
Theorem	le0neg1 10947	Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)
⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
Theorem	le0neg2 10948	Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)
⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ -𝐴 ≤ 0))
Theorem	addge01 10949	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵)))
Theorem	addge02 10950	A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐵 + 𝐴)))
Theorem	add20 10951	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 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
Theorem	subge0 10952	Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴))
Theorem	suble0 10953	Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 − 𝐵) ≤ 0 ↔ 𝐴 ≤ 𝐵))
Theorem	leaddle0 10954	The sum of a real number and a second real number is less than the real number iff the second real number is negative. (Contributed by Alexander van der Vekens, 30-May-2018.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) ≤ 𝐴 ↔ 𝐵 ≤ 0))
Theorem	subge02 10955	Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ (𝐴 − 𝐵) ≤ 𝐴))
Theorem	lesub0 10956	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 ≤ 𝐴 ∧ 𝐵 ≤ (𝐵 − 𝐴)) ↔ 𝐴 = 0))
Theorem	mulge0 10957	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 · 𝐵))
Theorem	mullt0 10958	The product of two negative numbers is positive. (Contributed by Jeff Hankins, 8-Jun-2009.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 𝐵 < 0)) → 0 < (𝐴 · 𝐵))
Theorem	msqgt0 10959	A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 6-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → 0 < (𝐴 · 𝐴))
Theorem	msqge0 10960	A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ (𝐴 ∈ ℝ → 0 ≤ (𝐴 · 𝐴))
Theorem	0lt1 10961	0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.)
⊢ 0 < 1
Theorem	0le1 10962	0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)
⊢ 0 ≤ 1
Theorem	relin01 10963	An interval law for less than or equal. (Contributed by Scott Fenton, 27-Jun-2013.)
⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ∨ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1) ∨ 1 ≤ 𝐴))
Theorem	ltordlem 10964*	Lemma for ltord1 10965. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 → 𝑀 < 𝑁))
Theorem	ltord1 10965*	Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑀 < 𝑁))
Theorem	leord1 10966*	Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑀 ≤ 𝑁))
Theorem	eqord1 10967*	A strictly increasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐴 < 𝐵))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁))
Theorem	ltord2 10968*	Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 < 𝐷 ↔ 𝑁 < 𝑀))
Theorem	leord2 10969*	Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 ≤ 𝐷 ↔ 𝑁 ≤ 𝑀))
Theorem	eqord2 10970*	A strictly decreasing real function on a subset of ℝ is one-to-one. (Contributed by Mario Carneiro, 14-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ (𝑥 = 𝐶 → 𝐴 = 𝑀)    &   ⊢ (𝑥 = 𝐷 → 𝐴 = 𝑁)    &   ⊢ 𝑆 ⊆ ℝ    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 < 𝑦 → 𝐵 < 𝐴))    ⇒   ⊢ ((𝜑 ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (𝐶 = 𝐷 ↔ 𝑀 = 𝑁))
Theorem	wloglei 10971*	Form of wlogle 10972 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑆 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜃)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)
Theorem	wlogle 10972*	If the predicate 𝜒(𝑥, 𝑦) is symmetric under interchange of 𝑥, 𝑦, then "without loss of generality" we can assume that 𝑥 ≤ 𝑦. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)
⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝑧 = 𝑦 ∧ 𝑤 = 𝑥) → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑆 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝜒 ↔ 𝜃))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑥 ≤ 𝑦)) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝜒)
Theorem	leidi 10973	'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 𝐴 ≤ 𝐴
Theorem	gt0ne0i 10974	Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (0 < 𝐴 → 𝐴 ≠ 0)
Theorem	gt0ne0ii 10975	Positive implies nonzero. (Contributed by NM, 15-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 0 < 𝐴    ⇒   ⊢ 𝐴 ≠ 0
Theorem	msqgt0i 10976	A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ (𝐴 ≠ 0 → 0 < (𝐴 · 𝐴))
Theorem	msqge0i 10977	A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    ⇒   ⊢ 0 ≤ (𝐴 · 𝐴)
Theorem	addgt0i 10978	Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
Theorem	addge0i 10979	Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 + 𝐵))
Theorem	addgegt0i 10980	Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 < 𝐵) → 0 < (𝐴 + 𝐵))
Theorem	addgt0ii 10981	Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 0 < 𝐴    &   ⊢ 0 < 𝐵    ⇒   ⊢ 0 < (𝐴 + 𝐵)
Theorem	add20i 10982	Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → ((𝐴 + 𝐵) = 0 ↔ (𝐴 = 0 ∧ 𝐵 = 0)))
Theorem	ltnegi 10983	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 ↔ -𝐵 < -𝐴)
Theorem	lenegi 10984	Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 ≤ 𝐵 ↔ -𝐵 ≤ -𝐴)
Theorem	ltnegcon2i 10985	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < -𝐵 ↔ 𝐵 < -𝐴)
Theorem	mulge0i 10986	The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 0 ≤ 𝐵) → 0 ≤ (𝐴 · 𝐵))
Theorem	lesub0i 10987	Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ ((0 ≤ 𝐴 ∧ 𝐵 ≤ (𝐵 − 𝐴)) ↔ 𝐴 = 0)
Theorem	ltaddposi 10988	Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))
Theorem	posdifi 10989	Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))
Theorem	ltnegcon1i 10990	Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (-𝐴 < 𝐵 ↔ -𝐵 < 𝐴)
Theorem	lenegcon1i 10991	Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)
Theorem	subge0i 10992	Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    ⇒   ⊢ (0 ≤ (𝐴 − 𝐵) ↔ 𝐵 ≤ 𝐴)
Theorem	ltadd1i 10993	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))
Theorem	leadd1i 10994	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (𝐴 ≤ 𝐵 ↔ (𝐴 + 𝐶) ≤ (𝐵 + 𝐶))
Theorem	leadd2i 10995	Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ (𝐴 ≤ 𝐵 ↔ (𝐶 + 𝐴) ≤ (𝐶 + 𝐵))
Theorem	ltsubaddi 10996	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 + 𝐵))
Theorem	lesubaddi 10997	'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 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐶 + 𝐵))
Theorem	ltsubadd2i 10998	'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 − 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶))
Theorem	lesubadd2i 10999	'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 − 𝐵) ≤ 𝐶 ↔ 𝐴 ≤ (𝐵 + 𝐶))
Theorem	ltaddsubi 11000	'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    ⇒   ⊢ ((𝐴 + 𝐵) < 𝐶 ↔ 𝐴 < (𝐶 − 𝐵))