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**Statement List for Metamath Proof Explorer -** 5601-5700 - Page 57 of 108
Type	Label	Description
Statement

Theorem	ltnsym 5601	'Less than' is not symmetric.


Theorem	lenlt 5602	'Less than or equal to' in terms of 'less than'.


Theorem	ltnle 5603	'Less than' in terms of 'less than or equal to'.


Theorem	ltle 5604	'Less than' implies 'less than or equal to'.


Theorem	ltlei 5605	'Less than' implies 'less than or equal to' (inference).


Theorem	eqle 5606	Equality implies 'less than or equal to'.


Theorem	ltne 5607	'Less than' implies not equal.


Theorem	letri 5608	Trichotomy law for 'less than or equal to'.
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Theorem	lttr 5609	'Less than' is transitive. Theorem I.17 of [Apostol] p. 20.
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Theorem	lelttr 5610	'Less than or equal to', 'less than' transitive law.
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Theorem	ltletr 5611	'Less than', 'less than or equal to' transitive law.
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Theorem	letr 5612	'Less than or equal to' is transitive.
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Theorem	le2tri3 5613	Extended trichotomy law for 'less than or equal to'.
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Theorem	ltadd2 5614	Addition to both sides of 'less than'. (Proof shortened by Paul Chapman, 27-Jan-2008.)


Theorem	ltadd1 5615	Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20.


Theorem	leadd1 5616	Addition to both sides of 'less than or equal to'.


Theorem	leadd2 5617	Addition to both sides of 'less than or equal to'.


Theorem	ltsubadd 5618	'Less than' relationship between subtraction and addition.


Theorem	lesubadd 5619	'Less than or equal to' relationship between subtraction and addition.


Theorem	lt2add 5620	Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20.
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Theorem	le2add 5621	Adding both side of two inequalities.
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Theorem	addgt0 5622	Addition of 2 positive numbers is positive.
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Theorem	addge0 5623	Addition of 2 nonnegative numbers is nonnegative.
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Theorem	addgegt0 5624	Addition of nonnegative and positive numbers is positive.
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Theorem	addgt0i 5625	Addition of 2 positive numbers is positive.


Theorem	add20 5626	Two nonnegative numbers are zero iff their sum is zero.
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Theorem	ltneg 5627	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.


Theorem	leneg 5628	Negative of both sides of 'less than or equal to'.


Theorem	ltnegcon2 5629	Contraposition of negative in 'less than'.


Theorem	mulgt0 5630	The product of two positive numbers is positive.
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Theorem	mulge0 5631	The product of two nonnegative numbers is nonnegative.
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Theorem	mulgt0i 5632	The product of two positive numbers is positive.


Theorem	ltnr 5633	'Less than' is irreflexive.


Theorem	leid 5634	'Less than or equal to' is reflexive.


Theorem	gt0ne0 5635	Positive means non-zero (useful for ordering theorems involving division).


Theorem	lesub0 5636	Lemma to show a nonnegative number is zero.
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Theorem	msqgt0 5637	A non-zero square is positive. Theorem I.20 of [Apostol] p. 20.


Theorem	msqge0 5638	A square is nonnegative.


Theorem	msqgt0t 5639	A non-zero square is positive. Theorem I.20 of [Apostol] p. 20.
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Theorem	msqge0t 5640	A square is nonnegative.


Theorem	gt0ne0i 5641	Positive implies nonzero.


Theorem	gt0ne0t 5642	Positive implies nonzero.
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Theorem	ne0gt0t 5643	A nonzero nonnegative number is positive.
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Theorem	letrit 5644	Trichotomy law.
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Theorem	lecase 5645	Ordering elimination by cases.
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Theorem	lelttrit 5646	Trichotomy law.
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Theorem	ltadd1t 5647	Addition to both sides of 'less than'.
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Theorem	ltadd2t 5648	Addition to both sides of 'less than'.
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Theorem	leadd1t 5649	Addition to both sides of 'less than or equal to'.
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Theorem	leadd2t 5650	Addition to both sides of 'less than or equal to'.
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Theorem	ltsubaddt 5651	'Less than' relationship between subtraction and addition.
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Theorem	ltsubadd2t 5652	'Less than' relationship between subtraction and addition.
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Theorem	lesubaddt 5653	'Less than or equal to' relationship between subtraction and addition.
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Theorem	lesubadd2t 5654	'Less than or equal to' relationship between subtraction and addition.
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Theorem	ltaddsubt 5655	'Less than' relationship between addition and subtraction.
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Theorem	ltaddsub2t 5656	'Less than' relationship between addition and subtraction.
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