mmtheorems57 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5601-5700 - Page 57 of 107

Type	Label	Description
Statement
Theorem	mulgt0i 5601	The product of two positive numbers is positive.
|- A e. RR   &   |- B e. RR   &   |- 0 < A   &   |- 0 < B   =>   |- 0 < (A x. B)
Theorem	ltnr 5602	'Less than' is irreflexive.
|- A e. RR   =>   |- -. A < A
Theorem	leid 5603	'Less than or equal to' is reflexive.
|- A e. RR   =>   |- A <_ A
Theorem	gt0ne0 5604	Positive means non-zero (useful for ordering theorems involving division).
|- A e. RR   =>   |- (0 < A -> A =/= 0)
Theorem	lesub0 5605	Lemma to show a nonnegative number is zero.
|- A e. RR   &   |- B e. RR   =>   |- ((0 <_ A /\ B <_ (B - A)) <-> A = 0)
Theorem	msqgt0 5606	A non-zero square is positive. Theorem I.20 of [Apostol] p. 20.
|- A e. RR   =>   |- (A =/= 0 -> 0 < (A x. A))
Theorem	msqge0 5607	A square is nonnegative.
|- A e. RR   =>   |- 0 <_ (A x. A)
Theorem	msqgt0t 5608	A non-zero square is positive. Theorem I.20 of [Apostol] p. 20.
|- ((A e. RR /\ A =/= 0) -> 0 < (A x. A))
Theorem	msqge0t 5609	A square is nonnegative.
|- (A e. RR -> 0 <_ (A x. A))
Theorem	gt0ne0i 5610	Positive implies nonzero.
|- A e. RR   &   |- 0 < A   =>   |- A =/= 0
Theorem	gt0ne0t 5611	Positive implies nonzero.
|- ((A e. RR /\ 0 < A) -> A =/= 0)
Theorem	ne0gt0t 5612	A nonzero nonnegative number is positive.
|- ((A e. RR /\ 0 <_ A) -> (A =/= 0 <-> 0 < A))
Theorem	letrit 5613	Trichotomy law.
|- ((A e. RR /\ B e. RR) -> (A <_ B \/ B <_ A))
Theorem	lecase 5614	Ordering elimination by cases.
|- (ph -> A e. RR)   &   |- (ph -> B e. RR)   &   |- ((ph /\ A <_ B) -> ps)   &   |- ((ph /\ B <_ A) -> ps)   =>   |- (ph -> ps)
Theorem	lelttrit 5615	Trichotomy law.
|- ((A e. RR /\ B e. RR) -> (A <_ B \/ B < A))
Theorem	ltadd1t 5616	Addition to both sides of 'less than'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < B <-> (A + C) < (B + C)))
Theorem	ltadd2t 5617	Addition to both sides of 'less than'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < B <-> (C + A) < (C + B)))
Theorem	leadd1t 5618	Addition to both sides of 'less than or equal to'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ B <-> (A + C) <_ (B + C)))
Theorem	leadd2t 5619	Addition to both sides of 'less than or equal to'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ B <-> (C + A) <_ (C + B)))
Theorem	ltsubaddt 5620	'Less than' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) < C <-> A < (C + B)))
Theorem	ltsubadd2t 5621	'Less than' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) < C <-> A < (B + C)))
Theorem	lesubaddt 5622	'Less than or equal to' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) <_ C <-> A <_ (C + B)))
Theorem	lesubadd2t 5623	'Less than or equal to' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) <_ C <-> A <_ (B + C)))
Theorem	ltaddsubt 5624	'Less than' relationship between addition and subtraction.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A + B) < C <-> A < (C - B)))
Theorem	ltaddsub2t 5625	'Less than' relationship between addition and subtraction.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A + B) < C <-> B < (C - A)))
Theorem	leaddsubt 5626	'Less than or equal to' relationship between addition and subtraction.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A + B) <_ C <-> A <_ (C - B)))
Theorem	leaddsub2t 5627	'Less than or equal to' relationship between and addition and subtraction.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A + B) <_ C <-> B <_ (C - A)))
Theorem	sublet 5628	Swap subtrahends in an inequality.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) <_ C <-> (A - C) <_ B))
Theorem	lesubt 5629	Swap subtrahends in an inequality.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ (B - C) <-> C <_ (B - A)))
Theorem	ltsubadd2 5630	'Less than' relationship between subtraction and addition.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((A - B) < C <-> A < (B + C))
Theorem	lesubadd2 5631	'Less than or equal to' relationship between subtraction and addition.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((A - B) <_ C <-> A <_ (B + C))
Theorem	ltaddsub 5632	'Less than' relationship between subtraction and addition.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- ((A + B) < C <-> A < (C - B))
Theorem	ltmullem 5633	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20.
|- A e. RR   &   |- B e. RR   &   |- C e. RR   =>   |- (0 < C -> (A < B -> (A x. C) < (B x. C)))
Theorem	ltsub23t 5634	'Less than' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> ((A - B) < C <-> (A - C) < B))
Theorem	ltsub13t 5635	'Less than' relationship between subtraction and addition.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < (B - C) <-> C < (B - A)))
Theorem	lt2addt 5636	Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A < C /\ B < D) -> (A + B) < (C + D)))
Theorem	le2addt 5637	Adding both sides of two 'less than or equal to' relations.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A <_ C /\ B <_ D) -> (A + B) <_ (C + D)))
Theorem	ltleaddt 5638	Adding both sides of two orderings.
|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A < C /\ B <_ D) -> (A + B) < (C + D)))
Theorem	addgt0t 5639	The sum of 2 positive numbers is positive.
|- (((A e. RR /\ B e. RR) /\ (0 < A /\ 0 < B)) -> 0 < (A + B))
Theorem	addgegt0t 5640	The sum of nonnegative and positive numbers is positive.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 < B)) -> 0 < (A + B))
Theorem	addgtge0t 5641	The sum of nonnegative and positive numbers is positive.
|- (((A e. RR /\ B e. RR) /\ (0 < A /\ 0 <_ B)) -> 0 < (A + B))
Theorem	addge0t 5642	The sum of 2 nonnegative numbers is nonnegative.
|- (((A e. RR /\ B e. RR) /\ (0 <_ A /\ 0 <_ B)) -> 0 <_ (A + B))
Theorem	ltaddpost 5643	Adding a positive number to another number increases it.
|- ((A e. RR /\ B e. RR) -> (0 < A <-> B < (B + A)))
Theorem	ltaddpos2t 5644	Adding a positive number to another number increases it.
|- ((A e. RR /\ B e. RR) -> (0 < A <-> B < (A + B)))
Theorem	ltsubpost 5645	Subtracting a positive number from another number decreases it.
|- ((A e. RR /\ B e. RR) -> (0 < A <-> (B - A) < B))
Theorem	posdift 5646	Comparison of two numbers whose difference is positive.
|- ((A e. RR /\ B e. RR) -> (A < B <-> 0 < (B - A)))
Theorem	ltnegt 5647	Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20.
|- ((A e. RR /\ B e. RR) -> (A < B <-> -uB < -uA))
Theorem	ltnegcon1t 5648	Contraposition of negative in 'less than'.
|- ((A e. RR /\ B e. RR) -> (-uA < B <-> -uB < A))
Theorem	lenegt 5649	Negative of both sides of 'less than or equal to'.
|- ((A e. RR /\ B e. RR) -> (A <_ B <-> -uB <_ -uA))
Theorem	lenegcon1t 5650	Contraposition of negative in 'less than or equal to'.
|- ((A e. RR /\ B e. RR) -> (-uA <_ B <-> -uB <_ A))
Theorem	lenegcon2t 5651	Contraposition of negative in 'less than or equal to'.
|- ((A e. RR /\ B e. RR) -> (A <_ -uB <-> B <_ -uA))
Theorem	lesub1t 5652	Subtraction from both sides of 'less than or equal to'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ B <-> (A - C) <_ (B - C)))
Theorem	lesub2t 5653	Subtraction of both sides of 'less than or equal to'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A <_ B <-> (C - B) <_ (C - A)))
Theorem	ltsub1t 5654	Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.)
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A < B <-> (A - C) < (B - C)))
Theorem	ltsub2t 5655	Subtraction of both sides of 'less than'.
|- ((A e. RR /\ B e. RR /\ C e. RR) -> (A