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
4.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 9401	Extend class notation to include the class of nonnegative integers.
class NN0
Definition	df-n0 9402	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 = ( NN u. { 0 } )
Theorem	elnn0 9403	Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
|- ( A e. NN0 <-> ( A e. NN \/ A = 0 ) )
Theorem	nnssnn0 9404	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN C_ NN0
Theorem	nn0ssre 9405	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 C_ RR
Theorem	nn0sscn 9406	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
|- NN0 C_ CC
Theorem	nn0ex 9407	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
|- NN0 e. _V
Theorem	nnnn0 9408	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
|- ( A e. NN -> A e. NN0 )
Theorem	nnnn0i 9409	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
|- N e. NN   =>    |- N e. NN0
Theorem	nn0re 9410	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. RR )
Theorem	nn0cn 9411	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. CC )
Theorem	nn0rei 9412	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. RR
Theorem	nn0cni 9413	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. CC
Theorem	dfn2 9414	The set of positive integers defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
|- NN = ( NN0 \ { 0 } )
Theorem	elnnne0 9415	The positive integer property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
|- ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
Theorem	0nn0 9416	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 0 e. NN0
Theorem	1nn0 9417	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 1 e. NN0
Theorem	2nn0 9418	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 2 e. NN0
Theorem	3nn0 9419	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 3 e. NN0
Theorem	4nn0 9420	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 4 e. NN0
Theorem	5nn0 9421	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 5 e. NN0
Theorem	6nn0 9422	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 6 e. NN0
Theorem	7nn0 9423	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 7 e. NN0
Theorem	8nn0 9424	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 8 e. NN0
Theorem	9nn0 9425	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 9 e. NN0
Theorem	nn0ge0 9426	A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
|- ( N e. NN0 -> 0 <_ N )
Theorem	nn0nlt0 9427	A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
|- ( A e. NN0 -> -. A < 0 )
Theorem	nn0ge0i 9428	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- 0 <_ N
Theorem	nn0le0eq0 9429	A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
|- ( N e. NN0 -> ( N <_ 0 <-> N = 0 ) )
Theorem	nn0p1gt0 9430	A nonnegative integer increased by 1 is greater than 0. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
|- ( N e. NN0 -> 0 < ( N + 1 ) )
Theorem	nnnn0addcl 9431	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 9432	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 9433	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 9434	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 9435	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 9436	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 9437	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 9438	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 9439	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 9440	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 9441	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 9442	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 9443	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 9444	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 9445	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 9446	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 9447	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 9448	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 9449	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 9450	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 9451	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 9452	'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 9453	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 9454	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9455	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9456	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9457	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 9458	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 9459	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 9460	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 9461	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 9462	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 9463	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 8217.
Syntax	cxnn0 9464	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 9465	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 8217. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7318 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 9466	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 9467	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 9468	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 9469	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 9470	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 9471	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 9472	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 9473	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 9474	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 9475	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 9476	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 9477	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 9478	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 9479	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 9480	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 9481	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 9482	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 9483	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 9484	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR
Theorem	zssre 9485	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ RR
Theorem	zsscn 9486	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ CC
Theorem	zex 9487	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 9488	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 9489	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 9490	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 9491	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 9492	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 9493	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 9494	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 9495	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 9496	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 9497	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 9498	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 9499	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 9500	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ