P. 93 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	divgt0i 9201	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A / B ) )
Theorem	divge0i 9202	The ratio of nonnegative and positive numbers is nonnegative. (Contributed by NM, 12-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 < B ) -> 0 <_ ( A / B ) )
Theorem	ltreci 9203	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )
Theorem	lereci 9204	The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 < A /\ 0 < B ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
Theorem	lt2msqi 9205	The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 3-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A < B <-> ( A x. A ) < ( B x. B ) ) )
Theorem	le2msqi 9206	The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( A <_ B <-> ( A x. A ) <_ ( B x. B ) ) )
Theorem	msq11i 9207	The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999.)
|- A e. RR   &    |- B e. RR   =>    |- ( ( 0 <_ A /\ 0 <_ B ) -> ( ( A x. A ) = ( B x. B ) <-> A = B ) )
Theorem	divgt0i2i 9208	The ratio of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < B   =>    |- ( 0 < A -> 0 < ( A / B ) )
Theorem	ltrecii 9209	The reciprocal of both sides of 'less than'. (Contributed by NM, 15-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- ( A < B <-> ( 1 / B ) < ( 1 / A ) )
Theorem	divgt0ii 9210	The ratio of two positive numbers is positive. (Contributed by NM, 18-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- 0 < A   &    |- 0 < B   =>    |- 0 < ( A / B )
Theorem	ltmul1i 9211	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
Theorem	ltdiv1i 9212	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( A / C ) < ( B / C ) ) )
Theorem	ltmuldivi 9213	'Less than' relationship between division and multiplication. (Contributed by NM, 12-Oct-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( ( A x. C ) < B <-> A < ( B / C ) ) )
Theorem	ltmul2i 9214	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )
Theorem	lemul1i 9215	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) )
Theorem	lemul2i 9216	Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( 0 < C -> ( A <_ B <-> ( C x. A ) <_ ( C x. B ) ) )
Theorem	ltdiv23i 9217	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   =>    |- ( ( 0 < B /\ 0 < C ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )
Theorem	ltdiv23ii 9218	Swap denominator with other side of 'less than'. (Contributed by NM, 26-Sep-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < B   &    |- 0 < C   =>    |- ( ( A / B ) < C <-> ( A / C ) < B )
Theorem	ltmul1ii 9219	Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 16-May-1999.) (Proof shortened by Paul Chapman, 25-Jan-2008.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < C   =>    |- ( A < B <-> ( A x. C ) < ( B x. C ) )
Theorem	ltdiv1ii 9220	Division of both sides of 'less than' by a positive number. (Contributed by NM, 16-May-1999.)
|- A e. RR   &    |- B e. RR   &    |- C e. RR   &    |- 0 < C   =>    |- ( A < B <-> ( A / C ) < ( B / C ) )
Theorem	ltp1d 9221	A number is less than itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A < ( A + 1 ) )
Theorem	lep1d 9222	A number is less than or equal to itself plus 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> A <_ ( A + 1 ) )
Theorem	ltm1d 9223	A number minus 1 is less than itself. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A - 1 ) < A )
Theorem	lem1d 9224	A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A - 1 ) <_ A )
Theorem	recgt0d 9225	The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 0 < A )   =>    |- ( ph -> 0 < ( 1 / A ) )
Theorem	divgt0d 9226	The ratio of two positive numbers is positive. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 < A )   &    |- ( ph -> 0 < B )   =>    |- ( ph -> 0 < ( A / B ) )
Theorem	mulgt1d 9227	The product of two numbers greater than 1 is greater than 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> 1 < B )   =>    |- ( ph -> 1 < ( A x. B ) )
Theorem	lemulge11d 9228	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 1 <_ B )   =>    |- ( ph -> A <_ ( A x. B ) )
Theorem	lemulge12d 9229	Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 1 <_ B )   =>    |- ( ph -> A <_ ( B x. A ) )
Theorem	lemul1ad 9230	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( A x. C ) <_ ( B x. C ) )
Theorem	lemul2ad 9231	Multiplication of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   =>    |- ( ph -> ( C x. A ) <_ ( C x. B ) )
Theorem	ltmul12ad 9232	Comparison of product of two positive numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> A < B )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> C < D )   =>    |- ( ph -> ( A x. C ) < ( B x. D ) )
Theorem	lemul12ad 9233	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ C )   &    |- ( ph -> A <_ B )   &    |- ( ph -> C <_ D )   =>    |- ( ph -> ( A x. C ) <_ ( B x. D ) )
Theorem	lemul12bd 9234	Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> D e. RR )   &    |- ( ph -> 0 <_ A )   &    |- ( ph -> 0 <_ D )   &    |- ( ph -> A <_ B )   &    |- ( ph -> C <_ D )   =>    |- ( ph -> ( A x. C ) <_ ( B x. D ) )
Theorem	mulle0r 9235	Multiplying a nonnegative number by a nonpositive number yields a nonpositive number. (Contributed by Jim Kingdon, 28-Oct-2021.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ 0 /\ 0 <_ B ) ) -> ( A x. B ) <_ 0 )
4.3.10  Suprema
Theorem	lbreu 9236*	If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> E! x e. S A. y e. S x <_ y )
Theorem	lbcl 9237*	If a set of reals contains a lower bound, it contains a unique lower bound that belongs to the set. (Contributed by NM, 9-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> ( iota_ x e. S A. y e. S x <_ y ) e. S )
Theorem	lble 9238*	If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) -> ( iota_ x e. S A. y e. S x <_ y ) <_ A )
Theorem	lbinf 9239*	If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> inf ( S , RR , < ) = ( iota_ x e. S A. y e. S x <_ y ) )
Theorem	lbinfcl 9240*	If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y ) -> inf ( S , RR , < ) e. S )
Theorem	lbinfle 9241*	If a set of reals contains a lower bound, its infimum is less than or equal to all members of the set. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( ( S C_ RR /\ E. x e. S A. y e. S x <_ y /\ A e. S ) -> inf ( S , RR , < ) <_ A )
Theorem	suprubex 9242*	A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. A )   =>    |- ( ph -> B <_ sup ( A , RR , < ) )
Theorem	suprlubex 9243*	The supremum of a nonempty bounded set of reals is the least upper bound. (Contributed by Jim Kingdon, 19-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( B < sup ( A , RR , < ) <-> E. z e. A B < z ) )
Theorem	suprnubex 9244*	An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( -. B < sup ( A , RR , < ) <-> A. z e. A -. B < z ) )
Theorem	suprleubex 9245*	The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.)
|- ( ph -> E. x e. RR ( A. y e. A -. x < y /\ A. y e. RR ( y < x -> E. z e. A y < z ) ) )   &    |- ( ph -> A C_ RR )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( sup ( A , RR , < ) <_ B <-> A. z e. A z <_ B ) )
Theorem	negiso 9246	Negation is an order anti-isomorphism of the real numbers, which is its own inverse. (Contributed by Mario Carneiro, 24-Dec-2016.)
|- F = ( x e. RR |-> -u x )   =>    |- ( F Isom < , `' < ( RR , RR ) /\ `' F = F )
Theorem	dfinfre 9247*	The infimum of a set of reals A. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.)
|- ( A C_ RR -> inf ( A , RR , < ) = U. { x e. RR | ( A. y e. A x <_ y /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) } )
Theorem	sup3exmid 9248*	If any inhabited set of real numbers bounded from above has a supremum, excluded middle follows. (Contributed by Jim Kingdon, 2-Apr-2023.)
|- ( ( u C_ RR /\ E. w w e. u /\ E. x e. RR A. y e. u y <_ x ) -> E. x e. RR ( A. y e. u -. x < y /\ A. y e. RR ( y < x -> E. z e. u y < z ) ) )   =>    |- DECID ph
4.3.11  Imaginary and complex number properties
Theorem	crap0 9249	The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)
|- ( ( A e. RR /\ B e. RR ) -> ( ( A # 0 \/ B # 0 ) <-> ( A + ( _i x. B ) ) # 0 ) )
Theorem	creur 9250*	The real part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> E! x e. RR E. y e. RR A = ( x + ( _i x. y ) ) )
Theorem	creui 9251*	The imaginary part of a complex number is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. CC -> E! y e. RR E. x e. RR A = ( x + ( _i x. y ) ) )
Theorem	cju 9252*	The complex conjugate of a complex number is unique. (Contributed by Mario Carneiro, 6-Nov-2013.)
|- ( A e. CC -> E! x e. CC ( ( A + x ) e. RR /\ ( _i x. ( A - x ) ) e. RR ) )
4.3.12  Function operation analogue theorems
Theorem	ofnegsub 9253	Function analogue of negsub 8537. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )
4.4  Integer sets
4.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 9254	Extend class notation to include the class of positive integers.
class NN
Definition	df-inn 9255*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 9256 instead. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.) (New usage is discouraged.)
|- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
Theorem	dfnn2 9256*	Definition of the set of positive integers. Another name for df-inn 9255. (Contributed by Jeff Hankins, 12-Sep-2013.) (Revised by Mario Carneiro, 3-May-2014.)
|- NN = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
Theorem	peano5nni 9257*	Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( ( 1 e. A /\ A. x e. A ( x + 1 ) e. A ) -> NN C_ A )
Theorem	nnssre 9258	The positive integers are a subset of the reals. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
|- NN C_ RR
Theorem	nnsscn 9259	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- NN C_ CC
Theorem	nnex 9260	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- NN e. _V
Theorem	nnre 9261	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. RR )
Theorem	nncn 9262	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. CC )
Theorem	nnrei 9263	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. RR
Theorem	nncni 9264	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. CC
Theorem	1nn 9265	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
|- 1 e. NN
Theorem	peano2nn 9266	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by NM, 11-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ( A e. NN -> ( A + 1 ) e. NN )
Theorem	nnred 9267	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR )
Theorem	nncnd 9268	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. CC )
Theorem	peano2nnd 9269	Peano postulate: a successor of a positive integer is a positive integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> ( A + 1 ) e. NN )
4.4.2  Principle of mathematical induction
Theorem	nnind 9270*	Principle of Mathematical Induction (inference schema). The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. See nnaddcl 9274 for an example of its use. This is an alternative for Metamath 100 proof #74. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 16-Jun-2013.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   &    |- ps   &    |- ( y e. NN -> ( ch -> th ) )   =>    |- ( A e. NN -> ta )
Theorem	nnindALT 9271*	Principle of Mathematical Induction (inference schema). The last four hypotheses give us the substitution instances we need; the first two are the induction step and the basis. This ALT version of nnind 9270 has a different hypothesis order. It may be easier to use with the metamath program's Proof Assistant, because "MM-PA> assign last" will be applied to the substitution instances first. We may eventually use this one as the official version. You may use either version. After the proof is complete, the ALT version can be changed to the non-ALT version with "MM-PA> minimize nnind /allow". (Contributed by NM, 7-Dec-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
|- ( y e. NN -> ( ch -> th ) )   &    |- ps   &    |- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = y -> ( ph <-> ch ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> th ) )   &    |- ( x = A -> ( ph <-> ta ) )   =>    |- ( A e. NN -> ta )
Theorem	nn1m1nn 9272	Every positive integer is one or a successor. (Contributed by Mario Carneiro, 16-May-2014.)
|- ( A e. NN -> ( A = 1 \/ ( A - 1 ) e. NN ) )
Theorem	nn1suc 9273*	If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( x = 1 -> ( ph <-> ps ) )   &    |- ( x = ( y + 1 ) -> ( ph <-> ch ) )   &    |- ( x = A -> ( ph <-> th ) )   &    |- ps   &    |- ( y e. NN -> ch )   =>    |- ( A e. NN -> th )
Theorem	nnaddcl 9274	Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.)
|- ( ( A e. NN /\ B e. NN ) -> ( A + B ) e. NN )
Theorem	nnmulcl 9275	Closure of multiplication of positive integers. (Contributed by NM, 12-Jan-1997.)
|- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) e. NN )
Theorem	nnmulcli 9276	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- A e. NN   &    |- B e. NN   =>    |- ( A x. B ) e. NN
Theorem	nnge1 9277	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 9278	A positive integer is less than or equal to one iff it is equal to one. (Contributed by NM, 3-Apr-2005.)
|- ( A e. NN -> ( A <_ 1 <-> A = 1 ) )
Theorem	nngt0 9279	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 9280	A positive integer is not less than one. (Contributed by NM, 18-Jan-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. NN -> -. A < 1 )
Theorem	0nnn 9281	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 9282	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 9283	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 9284	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 9285	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 9286	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 9287*	There exists a positive integer greater than or equal to any two others. (Contributed by NM, 18-Aug-1999.)
|- ( ( A e. NN /\ B e. NN ) -> E. x e. NN ( A <_ x /\ B <_ x ) )
Theorem	nn1gt1 9288	A positive integer is either one or greater than one. This is for NN; 0elnn 4746 is a similar theorem for om (the natural numbers as ordinals). (Contributed by Jim Kingdon, 7-Mar-2020.)
|- ( A e. NN -> ( A = 1 \/ 1 < A ) )
Theorem	nngt1ne1 9289	A positive integer is greater than one iff it is not equal to one. (Contributed by NM, 7-Oct-2004.)
|- ( A e. NN -> ( 1 < A <-> A =/= 1 ) )
Theorem	nndivre 9290	The quotient of a real and a positive integer is real. (Contributed by NM, 28-Nov-2008.)
|- ( ( A e. RR /\ N e. NN ) -> ( A / N ) e. RR )
Theorem	nnrecre 9291	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 9292	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 9293	Subtraction of positive integers. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( B - A ) e. NN ) )
Theorem	nnsubi 9294	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 9295*	Two ways to express " A divides B " for positive integers. (Contributed by NM, 3-Feb-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( E. x e. NN ( A x. x ) = B <-> ( B / A ) e. NN ) )
Theorem	nndivtr 9296	Transitive property of divisibility: if A divides B and B divides C, then A divides C. Typically, C would be an integer, although the theorem holds for complex C. (Contributed by NM, 3-May-2005.)
|- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\ ( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN )
Theorem	nnge1d 9297	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 9298	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 9299	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 9300	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )