P. 91 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1nn 9001	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
|- 1 e. NN
Theorem	peano2nn 9002	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 9003	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR )
Theorem	nncnd 9004	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. CC )
Theorem	peano2nnd 9005	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 9006*	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 9010 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 9007*	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 9006 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 9008	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 9009*	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 9010	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 9011	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 9012	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 9013	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 9014	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 9015	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 9016	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 9017	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 9018	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 9019	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 9020	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 9021	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 9022	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 9023*	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 9024	A positive integer is either one or greater than one. This is for NN; 0elnn 4655 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 9025	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 9026	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 9027	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 9028	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 9029	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 9030	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 9031*	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 9032	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 9033	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 9034	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 9035	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 9036	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
Theorem	nnrecred 9037	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 9038	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 9039	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 9040	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 7886 through df-9 9056), 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 7886 and df-1 7887). 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 9041	Extend class notation to include the number 2.
class 2
Syntax	c3 9042	Extend class notation to include the number 3.
class 3
Syntax	c4 9043	Extend class notation to include the number 4.
class 4
Syntax	c5 9044	Extend class notation to include the number 5.
class 5
Syntax	c6 9045	Extend class notation to include the number 6.
class 6
Syntax	c7 9046	Extend class notation to include the number 7.
class 7
Syntax	c8 9047	Extend class notation to include the number 8.
class 8
Syntax	c9 9048	Extend class notation to include the number 9.
class 9
Definition	df-2 9049	Define the number 2. (Contributed by NM, 27-May-1999.)
|- 2 = ( 1 + 1 )
Definition	df-3 9050	Define the number 3. (Contributed by NM, 27-May-1999.)
|- 3 = ( 2 + 1 )
Definition	df-4 9051	Define the number 4. (Contributed by NM, 27-May-1999.)
|- 4 = ( 3 + 1 )
Definition	df-5 9052	Define the number 5. (Contributed by NM, 27-May-1999.)
|- 5 = ( 4 + 1 )
Definition	df-6 9053	Define the number 6. (Contributed by NM, 27-May-1999.)
|- 6 = ( 5 + 1 )
Definition	df-7 9054	Define the number 7. (Contributed by NM, 27-May-1999.)
|- 7 = ( 6 + 1 )
Definition	df-8 9055	Define the number 8. (Contributed by NM, 27-May-1999.)
|- 8 = ( 7 + 1 )
Definition	df-9 9056	Define the number 9. (Contributed by NM, 27-May-1999.)
|- 9 = ( 8 + 1 )
Theorem	0ne1 9057	0 =/= 1 (common case). See aso 1ap0 8617. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 1
Theorem	1ne0 9058	1 =/= 0. See aso 1ap0 8617. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 1 =/= 0
Theorem	1m1e0 9059	( 1 - 1 ) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 - 1 ) = 0
Theorem	2re 9060	The number 2 is real. (Contributed by NM, 27-May-1999.)
|- 2 e. RR
Theorem	2cn 9061	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
|- 2 e. CC
Theorem	2ex 9062	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 2 e. _V
Theorem	2cnd 9063	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 2 e. CC )
Theorem	3re 9064	The number 3 is real. (Contributed by NM, 27-May-1999.)
|- 3 e. RR
Theorem	3cn 9065	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
|- 3 e. CC
Theorem	3ex 9066	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. _V
Theorem	4re 9067	The number 4 is real. (Contributed by NM, 27-May-1999.)
|- 4 e. RR
Theorem	4cn 9068	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 4 e. CC
Theorem	5re 9069	The number 5 is real. (Contributed by NM, 27-May-1999.)
|- 5 e. RR
Theorem	5cn 9070	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 5 e. CC
Theorem	6re 9071	The number 6 is real. (Contributed by NM, 27-May-1999.)
|- 6 e. RR
Theorem	6cn 9072	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 9073	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9074	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9075	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9076	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9077	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9078	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9079	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9080	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9081	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9082	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9083	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9084	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 9085	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 9086	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 9087	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9088	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9089	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9090	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9091	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9092	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9093	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9094	The number 9 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 9
4.4.4  Some properties of specific numbers This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.
Theorem	neg1cn 9095	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9096	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9097	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9098	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9099	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9100	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1