P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0mulcl 9401	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 9402	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 9403	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 9404	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 9405	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 9406	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 9407	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 9408	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 9409	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 9410	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 9411	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 9412	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 9413	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 9414	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 9415	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 9416	'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 9417	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 9418	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9419	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9420	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9421	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 9422	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 9423	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 9424	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 9425	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 9426	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 9427	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 )
4.4.8  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 8181.
Syntax	cxnn0 9428	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 9429	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 8181. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7283 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 9430	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 9431	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 9432	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 9433	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 9434	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 9435	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 9436	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 9437	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 9438	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 9439	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 9440	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 9441	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 )
4.4.9  Integers (as a subset of complex numbers)
Syntax	cz 9442	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 9443	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 9444	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 9445	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 9446	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 9447	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 9448	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR
Theorem	zssre 9449	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ RR
Theorem	zsscn 9450	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ CC
Theorem	zex 9451	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 9452	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 9453	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 9454	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 9455	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 9456	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 9457	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 9458	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 9459	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 9460	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 9461	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 9462	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 9463	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 9464	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ
Theorem	elnnz1 9465	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 9466	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN = { x e. ZZ | 1 <_ x }
Theorem	nn0zrab 9467	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN0 = { x e. ZZ | 0 <_ x }
Theorem	1z 9468	One is an integer. (Contributed by NM, 10-May-2004.)
|- 1 e. ZZ
Theorem	1zzd 9469	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. ZZ )
Theorem	2z 9470	Two is an integer. (Contributed by NM, 10-May-2004.)
|- 2 e. ZZ
Theorem	3z 9471	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. ZZ
Theorem	4z 9472	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|- 4 e. ZZ
Theorem	znegcl 9473	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> -u N e. ZZ )
Theorem	neg1z 9474	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 9475	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 9476	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 9477	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 9478	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 9479	Lemma for zaddcl 9482. 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 9480	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 9481	Lemma for zaddcl 9482. 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 9482	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 9483	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 9484	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 9485	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 9486	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 9487	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 9488	'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 9489	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 9490	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	nnnle0 9491	A positive integer is not less than or equal to zero. (Contributed by AV, 13-May-2020.)
|- ( A e. NN -> -. A <_ 0 )
Theorem	zletr 9492	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 9493	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 9494	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9145.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 9495	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 ) )
Theorem	zmulcl 9496	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Theorem	zltp1le 9497	Integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	zleltp1 9498	Integer ordering relation. (Contributed by NM, 10-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	zlem1lt 9499	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	zltlem1 9500	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )