P. 94 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	indfval 9301	Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
|- ( ( O e. V /\ A C_ O /\ X e. O ) -> ( ( (𝟭 ` O ) ` A ) ` X ) = if ( X e. A , 1 , 0 ) )
Theorem	ind1 9302	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
|- ( ( O e. V /\ A C_ O /\ X e. A ) -> ( ( (𝟭 ` O ) ` A ) ` X ) = 1 )
Theorem	ind0 9303	Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
|- ( ( O e. V /\ A C_ O /\ X e. ( O \ A ) ) -> ( ( (𝟭 ` O ) ` A ) ` X ) = 0 )
Theorem	indconst0 9304	Indicator of the empty set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
|- ( O e. V -> ( (𝟭 ` O ) ` (/) ) = ( O X. { 0 } ) )
Theorem	indconst1 9305	Indicator of the whole set. (Contributed by Thierry Arnoux, 25-Jan-2026.)
|- ( O e. V -> ( (𝟭 ` O ) ` O ) = ( O X. { 1 } ) )
4.4  Integer sets
4.4.1  Positive integers (as a subset of complex numbers)
Syntax	cn 9306	Extend class notation to include the class of positive integers.
class NN
Definition	df-inn 9307*	Definition of the set of positive integers. For naming consistency with the Metamath Proof Explorer usages should refer to dfnn2 9308 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 9308*	Definition of the set of positive integers. Another name for df-inn 9307. (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 9309*	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 9310	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 9311	The positive integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- NN C_ CC
Theorem	nnex 9312	The set of positive integers exists. (Contributed by NM, 3-Oct-1999.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- NN e. _V
Theorem	nnre 9313	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. RR )
Theorem	nncn 9314	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- ( A e. NN -> A e. CC )
Theorem	nnrei 9315	A positive integer is a real number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. RR
Theorem	nncni 9316	A positive integer is a complex number. (Contributed by NM, 18-Aug-1999.)
|- A e. NN   =>    |- A e. CC
Theorem	1nn 9317	Peano postulate: 1 is a positive integer. (Contributed by NM, 11-Jan-1997.)
|- 1 e. NN
Theorem	peano2nn 9318	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 9319	A positive integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. RR )
Theorem	nncnd 9320	A positive integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. CC )
Theorem	peano2nnd 9321	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 9322*	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 9326 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 9323*	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 9322 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 9324	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 9325*	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 9326	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 9327	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 9328	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 9329	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 9330	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 9331	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 9332	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 9333	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 9334	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 9335	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 9336	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 9337	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 9338	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 9339*	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 9340	A positive integer is either one or greater than one. This is for NN; 0elnn 4766 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 9341	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 9342	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 9343	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 9344	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 9345	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 9346	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 9347*	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 9348	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 9349	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 9350	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 9351	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 9352	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
Theorem	nnrecred 9353	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 9354	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 9355	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 9356	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 8186 through df-9 9372), 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 8186 and df-1 8187). 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 9357	Extend class notation to include the number 2.
class 2
Syntax	c3 9358	Extend class notation to include the number 3.
class 3
Syntax	c4 9359	Extend class notation to include the number 4.
class 4
Syntax	c5 9360	Extend class notation to include the number 5.
class 5
Syntax	c6 9361	Extend class notation to include the number 6.
class 6
Syntax	c7 9362	Extend class notation to include the number 7.
class 7
Syntax	c8 9363	Extend class notation to include the number 8.
class 8
Syntax	c9 9364	Extend class notation to include the number 9.
class 9
Definition	df-2 9365	Define the number 2. (Contributed by NM, 27-May-1999.)
|- 2 = ( 1 + 1 )
Definition	df-3 9366	Define the number 3. (Contributed by NM, 27-May-1999.)
|- 3 = ( 2 + 1 )
Definition	df-4 9367	Define the number 4. (Contributed by NM, 27-May-1999.)
|- 4 = ( 3 + 1 )
Definition	df-5 9368	Define the number 5. (Contributed by NM, 27-May-1999.)
|- 5 = ( 4 + 1 )
Definition	df-6 9369	Define the number 6. (Contributed by NM, 27-May-1999.)
|- 6 = ( 5 + 1 )
Definition	df-7 9370	Define the number 7. (Contributed by NM, 27-May-1999.)
|- 7 = ( 6 + 1 )
Definition	df-8 9371	Define the number 8. (Contributed by NM, 27-May-1999.)
|- 8 = ( 7 + 1 )
Definition	df-9 9372	Define the number 9. (Contributed by NM, 27-May-1999.)
|- 9 = ( 8 + 1 )
Theorem	0ne1 9373	0 =/= 1 (common case). See aso 1ap0 8920. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 1
Theorem	1ne0 9374	1 =/= 0. See aso 1ap0 8920. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 1 =/= 0
Theorem	1m1e0 9375	( 1 - 1 ) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 - 1 ) = 0
Theorem	2re 9376	The number 2 is real. (Contributed by NM, 27-May-1999.)
|- 2 e. RR
Theorem	2cn 9377	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
|- 2 e. CC
Theorem	2ex 9378	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 2 e. _V
Theorem	2cnd 9379	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 2 e. CC )
Theorem	3re 9380	The number 3 is real. (Contributed by NM, 27-May-1999.)
|- 3 e. RR
Theorem	3cn 9381	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
|- 3 e. CC
Theorem	3ex 9382	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. _V
Theorem	4re 9383	The number 4 is real. (Contributed by NM, 27-May-1999.)
|- 4 e. RR
Theorem	4cn 9384	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 4 e. CC
Theorem	5re 9385	The number 5 is real. (Contributed by NM, 27-May-1999.)
|- 5 e. RR
Theorem	5cn 9386	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 5 e. CC
Theorem	6re 9387	The number 6 is real. (Contributed by NM, 27-May-1999.)
|- 6 e. RR
Theorem	6cn 9388	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 9389	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9390	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9391	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9392	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9393	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9394	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9395	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9396	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9397	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9398	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9399	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9400	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3