P. 89 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 8801-8900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nnnn0addcl 8801	A positive integer plus a nonnegative integer is a positive integer. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN /\ N e. NN0 ) -> ( M + N ) e. NN )
Theorem	nn0nnaddcl 8802	A nonnegative integer plus a positive integer is a positive integer. (Contributed by NM, 22-Dec-2005.)
|- ( ( M e. NN0 /\ N e. NN ) -> ( M + N ) e. NN )
Theorem	0mnnnnn0 8803	The result of subtracting a positive integer from 0 is not a nonnegative integer. (Contributed by Alexander van der Vekens, 19-Mar-2018.)
|- ( N e. NN -> ( 0 - N ) e/ NN0 )
Theorem	un0addcl 8804	If S is closed under addition, then so is S u. { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ph -> S C_ CC )   &    |- T = ( S u. { 0 } )   &    |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M + N ) e. S )   =>    |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T )
Theorem	un0mulcl 8805	If S is closed under multiplication, then so is S u. { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
|- ( ph -> S C_ CC )   &    |- T = ( S u. { 0 } )   &    |- ( ( ph /\ ( M e. S /\ N e. S ) ) -> ( M x. N ) e. S )   =>    |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T )
Theorem	nn0addcl 8806	Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M + N ) e. NN0 )
Theorem	nn0mulcl 8807	Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M x. N ) e. NN0 )
Theorem	nn0addcli 8808	Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
|- M e. NN0   &    |- N e. NN0   =>    |- ( M + N ) e. NN0
Theorem	nn0mulcli 8809	Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
|- M e. NN0   &    |- N e. NN0   =>    |- ( M x. N ) e. NN0
Theorem	nn0p1nn 8810	A nonnegative integer plus 1 is a positive integer. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( N e. NN0 -> ( N + 1 ) e. NN )
Theorem	peano2nn0 8811	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 8812	A positive integer minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( N e. NN -> ( N - 1 ) e. NN0 )
Theorem	elnn0nn 8813	The nonnegative integer property expressed in terms of positive integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( N e. NN0 <-> ( N e. CC /\ ( N + 1 ) e. NN ) )
Theorem	elnnnn0 8814	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
|- ( N e. NN <-> ( N e. CC /\ ( N - 1 ) e. NN0 ) )
Theorem	elnnnn0b 8815	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
|- ( N e. NN <-> ( N e. NN0 /\ 0 < N ) )
Theorem	elnnnn0c 8816	The positive integer property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
|- ( N e. NN <-> ( N e. NN0 /\ 1 <_ N ) )
Theorem	nn0addge1 8817	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( A + N ) )
Theorem	nn0addge2 8818	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
|- ( ( A e. RR /\ N e. NN0 ) -> A <_ ( N + A ) )
Theorem	nn0addge1i 8819	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
|- A e. RR   &    |- N e. NN0   =>    |- A <_ ( A + N )
Theorem	nn0addge2i 8820	A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
|- A e. RR   &    |- N e. NN0   =>    |- A <_ ( N + A )
Theorem	nn0le2xi 8821	A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- N <_ ( 2 x. N )
Theorem	nn0lele2xi 8822	'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- M e. NN0   &    |- N e. NN0   =>    |- ( N <_ M -> N <_ ( 2 x. M ) )
Theorem	nn0supp 8823	Two ways to write the support of a function on NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- ( F : I --> NN0 -> ( `' F " ( _V \ { 0 } ) ) = ( `' F " NN ) )
Theorem	nnnn0d 8824	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 8825	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 8826	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 8827	A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> 0 <_ A )
Theorem	nn0addcld 8828	Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   &    |- ( ph -> B e. NN0 )   =>    |- ( ph -> ( A + B ) e. NN0 )
Theorem	nn0mulcld 8829	Closure of multiplication of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   &    |- ( ph -> B e. NN0 )   =>    |- ( ph -> ( A x. B ) e. NN0 )
Theorem	nn0readdcl 8830	Closure law for addition of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) e. RR )
Theorem	nn0ge2m1nn 8831	If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is a positive integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 4-Jan-2020.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN )
Theorem	nn0ge2m1nn0 8832	If a nonnegative integer is greater than or equal to two, the integer decreased by 1 is also a nonnegative integer. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
|- ( ( N e. NN0 /\ 2 <_ N ) -> ( N - 1 ) e. NN0 )
Theorem	nn0nndivcl 8833	Closure law for dividing of a nonnegative integer by a positive integer. (Contributed by Alexander van der Vekens, 14-Apr-2018.)
|- ( ( K e. NN0 /\ L e. NN ) -> ( K / L ) e. RR )
3.4.8  Extended nonnegative integers The function values of the hash (set size) function are either nonnegative integers or positive infinity. To avoid the need to distinguish between finite and infinite sets (and therefore if the set size is a nonnegative integer or positive infinity), it is useful to provide a definition of the set of nonnegative integers extended by positive infinity, analogously to the extension of the real numbers RR*, see df-xr 7623.
Syntax	cxnn0 8834	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 8835	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 7623. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 6878 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 8836	An extended nonnegative integer is either a standard nonnegative integer or positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
Theorem	nn0ssxnn0 8837	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 8838	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 8839	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 8840	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 8841	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 8842	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 8843	A standard nonnegative integer is an extended nonnegative integer, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. NN0* )
Theorem	nn0nepnfd 8844	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 8845	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 8846	The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
Theorem	xnn0nnn0pnf 8847	An extended nonnegative integer which is not a standard nonnegative integer is positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( ( N e. NN0* /\ -. N e. NN0 ) -> N = +oo )
3.4.9  Integers (as a subset of complex numbers)
Syntax	cz 8848	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 8849	Define the set of integers, which are the positive and negative integers together with zero. Definition of integers in [Apostol] p. 22. The letter Z abbreviates the German word Zahlen meaning "numbers." (Contributed by NM, 8-Jan-2002.)
|- ZZ = { n e. RR | ( n = 0 \/ n e. NN \/ -u n e. NN ) }
Theorem	elz 8850	Membership in the set of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
Theorem	nnnegz 8851	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 8852	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 8853	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 8854	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR
Theorem	zssre 8855	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ RR
Theorem	zsscn 8856	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ CC
Theorem	zex 8857	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 8858	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 8859	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 8860	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 8861	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 8862	Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.)
|- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
Theorem	elznn0 8863	Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) )
Theorem	elznn 8864	Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005.)
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )
Theorem	nnssz 8865	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 8866	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 8867	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 8868	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 8869	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 8870	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ
Theorem	elnnz1 8871	Positive integer property expressed in terms of integers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( N e. NN <-> ( N e. ZZ /\ 1 <_ N ) )
Theorem	nnzrab 8872	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN = { x e. ZZ | 1 <_ x }
Theorem	nn0zrab 8873	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN0 = { x e. ZZ | 0 <_ x }
Theorem	1z 8874	One is an integer. (Contributed by NM, 10-May-2004.)
|- 1 e. ZZ
Theorem	1zzd 8875	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. ZZ )
Theorem	2z 8876	Two is an integer. (Contributed by NM, 10-May-2004.)
|- 2 e. ZZ
Theorem	3z 8877	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. ZZ
Theorem	4z 8878	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|- 4 e. ZZ
Theorem	znegcl 8879	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> -u N e. ZZ )
Theorem	neg1z 8880	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 8881	A number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014.)
|- ( A e. CC -> ( A e. ZZ <-> -u A e. ZZ ) )
Theorem	nn0negz 8882	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 8883	The negative of a nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- -u N e. ZZ
Theorem	peano2z 8884	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 8885	Lemma for zaddcl 8888. Special case in which N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. NN ) -> ( M + N ) e. ZZ )
Theorem	peano2zm 8886	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 8887	Lemma for zaddcl 8888. Special case in which -u N is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. RR /\ -u N e. NN ) -> ( M + N ) e. ZZ )
Theorem	zaddcl 8888	Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + N ) e. ZZ )
Theorem	zsubcl 8889	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 8890	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 8891	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 8892	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 8893	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 8894	'Less than' expressed in terms of 'less than or equal to'. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A < B <-> -. B <_ A ) )
Theorem	zleloe 8895	Integer 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by Jim Kingdon, 8-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
Theorem	znnnlt1 8896	An integer is not a positive integer iff it is less than one. (Contributed by NM, 13-Jul-2005.)
|- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) )
Theorem	zletr 8897	Transitive law of ordering for integers. (Contributed by Alexander van der Vekens, 3-Apr-2018.)
|- ( ( J e. ZZ /\ K e. ZZ /\ L e. ZZ ) -> ( ( J <_ K /\ K <_ L ) -> J <_ L ) )
Theorem	zrevaddcl 8898	Reverse closure law for addition of integers. (Contributed by NM, 11-May-2004.)
|- ( N e. ZZ -> ( ( M e. CC /\ ( M + N ) e. ZZ ) <-> M e. ZZ ) )
Theorem	znnsub 8899	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 8559.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 8900	The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( A + B ) e. ( RR \ ZZ ) )