Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 4301-4400 - Page 44 of 58
Type	Label	Description
Statement

Theorem	lelt 4301	'Less than or equal to' in terms of 'less than'.


Theorem	ltle 4302	'Less than' implies 'less than or equal to'.


Theorem	ltlei 4303	'Less than' implies 'less than or equal to' (inference).


Theorem	eqle 4304	Equality implies 'less than or equal to'.


Theorem	ltne 4305	'Less than' implies not equal.


Theorem	letri 4306	Trichotomy law for 'less than or equal to'.
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Theorem	lttr 4307	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20.
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Theorem	lelttr 4308	'Less than or equal to', 'less than' transitive law.
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Theorem	ltletr 4309	'Less than', 'less than or equal to' transitive law.
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Theorem	letr 4310	'Less than or equal to' is transitive.
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Theorem	le2tri3 4311	Extended trichotomy law for 'less than or equal to'.
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Theorem	ltadd2 4312	Addition to both sides of 'less than'.


Theorem	ltadd1 4313	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.


Theorem	leadd1 4314	Addition to both sides of 'less than or equal to'.


Theorem	leadd2 4315	Addition to both sides of 'less than or equal to'.


Theorem	ltsubadd 4316	'Less than' relationship between subtraction and addition.


Theorem	ltsubadd2 4317	'Less than' relationship between subtraction and addition.


Theorem	lesubadd2 4318	'Less than or equal to' relationship between subtraction and addition.


Theorem	lesubadd 4319	'Less than or equal to' relationship between subtraction and addition.


Theorem	ltaddsub 4320	'Less than' relationship between subtraction and addition.


Theorem	lt2add 4321	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20.
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Theorem	le2add 4322	Adding both side of two inequalities.
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Theorem	addgt0 4323	Addition of 2 positive numbers is positive.
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Theorem	addge0 4324	Addition of 2 nonnegative numbers is nonnegative.
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Theorem	addgegt0 4325	Addition of nonnegative and positive numbers is positive.
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Theorem	addgt0i 4326	Addition of 2 positive numbers is positive.


Theorem	ltaddpos 4327	Adding a positive number to another number increases it.


Theorem	posdif 4328	Comparison of two numbers whose difference is positive.


Theorem	add20 4329	Two nonnegative numbers are zero iff their sum is zero.
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Theorem	ltneg 4330	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.


Theorem	leneg 4331	Negative of both sides of 'less than or equal to'.


Theorem	ltnegcon1 4332	Contraposition of negative in 'less than'.


Theorem	ltnegcon2 4333	Contraposition of negative in 'less than'.


Theorem	mulgt0 4334	The product of two positive numbers is positive.
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Theorem	mulge0 4335	The product of two nonnegative numbers is nonnegative.
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Theorem	mulgt0i 4336	The product of two positive numbers is positive.


Theorem	ltmullem 4337	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.


Theorem	ltnr 4338	'Less than' is irreflexive.


Theorem	leid 4339	'Less than or equal to' is reflexive.


Theorem	gt0ne0 4340	Positive means non-zero (useful for ordering theorems involving division).


Theorem	lesub0 4341	Lemma to show a nonnegative number is zero.
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Theorem	subge0 4342	Nonnegative subtraction.


Theorem	sqgt0 4343	A non-zero square is positive. Theorem I.20 of [Apostol] p. 20.


Theorem	sqge0 4344	A square is nonnegative.


Theorem	gt0ne0i 4345	Positive implies nonzero.


Theorem	gt0ne0t 4346	Positive implies nonzero.


Theorem	letrit 4347	Trichotomy law.
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Theorem	ltadd1t 4348	Addition to both sides of 'less than'.
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Theorem	ltadd2t 4349	Addition to both sides of 'less than'.
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Theorem	leadd1t 4350	Addition to both sides of 'less than or equal to'.
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Theorem	leadd2t 4351	Addition to both sides of 'less than or equal to'.
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Theorem	lesub1t 4352	Subtraction from both sides of 'less than or equal to'.
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Theorem	ltsubaddt 4353	'Less than' relationship between subtraction and addition.
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Theorem	ltsubadd2t 4354	'Less than' relationship between subtraction and addition.
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Theorem	lesubaddt 4355	'Less than or equal to' relationship between subtraction and addition.
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Theorem	lesubadd2t 4356	'Less than or equal to' relationship between subtraction and addition.
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Theorem	ltaddsubt 4357	'Less than' relationship between subtraction and addition.
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Theorem	ltaddsub2t 4358	'Less than' relationship between subtraction and addition.
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