P. 86 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8501-8600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn1m1nn 8501	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 8502*	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 8503	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 8504	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 8505	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 8506	A positive integer is one or greater. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 1 <_ A )
Theorem	nnle1eq1 8507	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 8508	A positive integer is positive. (Contributed by NM, 26-Sep-1999.)
|- ( A e. NN -> 0 < A )
Theorem	nnnlt1 8509	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 8510	Zero is not a positive integer. (Contributed by NM, 25-Aug-1999.)
|- -. 0 e. NN
Theorem	nnne0 8511	A positive integer is nonzero. (Contributed by NM, 27-Sep-1999.)
|- ( A e. NN -> A =/= 0 )
Theorem	nnap0 8512	A positive integer is apart from zero. (Contributed by Jim Kingdon, 8-Mar-2020.)
|- ( A e. NN -> A # 0 )
Theorem	nngt0i 8513	A positive integer is positive (inference version). (Contributed by NM, 17-Sep-1999.)
|- A e. NN   =>    |- 0 < A
Theorem	nnap0i 8514	A positive integer is apart from zero (inference version). (Contributed by Jim Kingdon, 1-Jan-2023.)
|- A e. NN   =>    |- A # 0
Theorem	nnne0i 8515	A positive integer is nonzero (inference version). (Contributed by NM, 25-Aug-1999.)
|- A e. NN   =>    |- A =/= 0
Theorem	nn2ge 8516*	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 8517	A positive integer is either one or greater than one. This is for NN; 0elnn 4445 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 8518	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 8519	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 8520	The reciprocal of a positive integer is real. (Contributed by NM, 8-Feb-2008.)
|- ( N e. NN -> ( 1 / N ) e. RR )
Theorem	nnrecgt0 8521	The reciprocal of a positive integer is positive. (Contributed by NM, 25-Aug-1999.)
|- ( A e. NN -> 0 < ( 1 / A ) )
Theorem	nnsub 8522	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 8523	Subtraction of positive integers. (Contributed by NM, 19-Aug-2001.)
|- A e. NN   &    |- B e. NN   =>    |- ( A < B <-> ( B - A ) e. NN )
Theorem	nndiv 8524*	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 8525	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 8526	A positive integer is one or greater. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 1 <_ A )
Theorem	nngt0d 8527	A positive integer is positive. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> 0 < A )
Theorem	nnne0d 8528	A positive integer is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A =/= 0 )
Theorem	nnap0d 8529	A positive integer is apart from zero. (Contributed by Jim Kingdon, 25-Aug-2021.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A # 0 )
Theorem	nnrecred 8530	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 8531	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 8532	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 8533	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 )
3.4.3  Decimal representation of numbers The decimal representation of numbers/integers is based on the decimal digits 0 through 9 (df-0 7418 through df-9 8549), 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 7418 and df-1 7419). 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 8534	Extend class notation to include the number 2.
class 2
Syntax	c3 8535	Extend class notation to include the number 3.
class 3
Syntax	c4 8536	Extend class notation to include the number 4.
class 4
Syntax	c5 8537	Extend class notation to include the number 5.
class 5
Syntax	c6 8538	Extend class notation to include the number 6.
class 6
Syntax	c7 8539	Extend class notation to include the number 7.
class 7
Syntax	c8 8540	Extend class notation to include the number 8.
class 8
Syntax	c9 8541	Extend class notation to include the number 9.
class 9
Definition	df-2 8542	Define the number 2. (Contributed by NM, 27-May-1999.)
|- 2 = ( 1 + 1 )
Definition	df-3 8543	Define the number 3. (Contributed by NM, 27-May-1999.)
|- 3 = ( 2 + 1 )
Definition	df-4 8544	Define the number 4. (Contributed by NM, 27-May-1999.)
|- 4 = ( 3 + 1 )
Definition	df-5 8545	Define the number 5. (Contributed by NM, 27-May-1999.)
|- 5 = ( 4 + 1 )
Definition	df-6 8546	Define the number 6. (Contributed by NM, 27-May-1999.)
|- 6 = ( 5 + 1 )
Definition	df-7 8547	Define the number 7. (Contributed by NM, 27-May-1999.)
|- 7 = ( 6 + 1 )
Definition	df-8 8548	Define the number 8. (Contributed by NM, 27-May-1999.)
|- 8 = ( 7 + 1 )
Definition	df-9 8549	Define the number 9. (Contributed by NM, 27-May-1999.)
|- 9 = ( 8 + 1 )
Theorem	0ne1 8550	0 =/= 1 (common case). See aso 1ap0 8128. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 1
Theorem	1ne0 8551	1 =/= 0. See aso 1ap0 8128. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 1 =/= 0
Theorem	1m1e0 8552	( 1 - 1 ) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 - 1 ) = 0
Theorem	2re 8553	The number 2 is real. (Contributed by NM, 27-May-1999.)
|- 2 e. RR
Theorem	2cn 8554	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
|- 2 e. CC
Theorem	2ex 8555	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 2 e. _V
Theorem	2cnd 8556	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 2 e. CC )
Theorem	3re 8557	The number 3 is real. (Contributed by NM, 27-May-1999.)
|- 3 e. RR
Theorem	3cn 8558	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
|- 3 e. CC
Theorem	3ex 8559	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. _V
Theorem	4re 8560	The number 4 is real. (Contributed by NM, 27-May-1999.)
|- 4 e. RR
Theorem	4cn 8561	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 4 e. CC
Theorem	5re 8562	The number 5 is real. (Contributed by NM, 27-May-1999.)
|- 5 e. RR
Theorem	5cn 8563	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 5 e. CC
Theorem	6re 8564	The number 6 is real. (Contributed by NM, 27-May-1999.)
|- 6 e. RR
Theorem	6cn 8565	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 8566	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 8567	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 8568	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 8569	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 8570	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 8571	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 8572	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 8573	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 8574	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 8575	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 8576	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 8577	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 8578	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 8579	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 8580	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 8581	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 8582	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 8583	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 8584	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 8585	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 8586	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 8587	The number 9 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 9
3.4.4  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 8588	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 8589	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 8590	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 8591	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 8592	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 8593	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 8594	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 8595	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 8596	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 8597	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 8598	Function value at N + 1 with N replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )
Theorem	1p0e1 8599	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 8600	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2