P. 88 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8701-8800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	divgt0ii 8701	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 8702	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 8703	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 8704	'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 8705	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 8706	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 8707	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 8708	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 8709	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 8710	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 8711	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 8712	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 8713	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 8714	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 8715	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 8716	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 8717	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 8718	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 8719	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 8720	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 8721	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 8722	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 8723	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 8724	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 8725	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 8726	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 8727*	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 8728*	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 8729*	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 8730*	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 8731*	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 8732*	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 8733*	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 8734*	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 8735*	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 8736*	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 8737	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 8738*	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 8739*	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 8740	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 8741*	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 8742*	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 8743*	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 ) )
4.4  Integer sets
4.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 8744	Extend class notation to include the class of positive integers.
class NN
Definition	df-inn 8745*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 8746 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 8746*	Definition of the set of positive integers. Another name for df-inn 8745. (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 8747*	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 8748	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 8749	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- NN C_ CC
Theorem	nnex 8750	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- NN e. _V
Theorem	nnre 8751	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. RR )
Theorem	nncn 8752	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. CC )
Theorem	nnrei 8753	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. RR
Theorem	nncni 8754	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. CC
Theorem	1nn 8755	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
|- 1 e. NN
Theorem	peano2nn 8756	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 8757	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR )
Theorem	nncnd 8758	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. CC )
Theorem	peano2nnd 8759	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 8760*	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 8764 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 8761*	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 8760 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 8762	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 8763*	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 8764	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 8765	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 8766	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 8767	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 8768	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 8769	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 8770	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 8771	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 8772	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 8773	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 8774	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 8775	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 8776	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 8777*	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 8778	A positive integer is either one or greater than one. This is for NN; 0elnn 4540 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 8779	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 8780	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 8781	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 8782	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 8783	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 8784	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 8785*	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 8786	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 8787	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 8788	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 8789	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 8790	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
Theorem	nnrecred 8791	The reciprocal of a positive integer is real. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> ( 1 / A ) e. RR )
Theorem	nnaddcld 8792	Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A + B ) e. NN )
Theorem	nnmulcld 8793	Closure of multiplication of positive integers. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A x. B ) e. NN )
Theorem	nndivred 8794	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. NN )   =>    |- ( ph -> ( A / B ) e. RR )
4.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7651 through df-9 8810), which are explicitly defined in the following. Note that the numbers 0 and 1 are constants defined as primitives of the complex number axiom system (see df-0 7651 and df-1 7652). Integers can also be exhibited as sums of powers of 10 (e.g. the number 103 can be expressed as ( (; 1 0 ^ 2 ) + 3 )) or as some other expression built from operations on the numbers 0 through 9. For example, the prime number 823541 can be expressed as ( 7 ^ 7 ) - 2. Most abstract math rarely requires numbers larger than 4. Even in Wiles' proof of Fermat's Last Theorem, the largest number used appears to be 12.
Syntax	c2 8795	Extend class notation to include the number 2.
class 2
Syntax	c3 8796	Extend class notation to include the number 3.
class 3
Syntax	c4 8797	Extend class notation to include the number 4.
class 4
Syntax	c5 8798	Extend class notation to include the number 5.
class 5
Syntax	c6 8799	Extend class notation to include the number 6.
class 6
Syntax	c7 8800	Extend class notation to include the number 7.
class 7