P. 93 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9201-9300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0ge0 9201	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 9202	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 9203	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- 0 <_ N
Theorem	nn0le0eq0 9204	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 9205	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 9206	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 9207	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 9208	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 9209	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 9210	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 9211	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 9212	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 9213	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 9214	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 9215	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 9216	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 9217	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 9218	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 9219	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 9220	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 9221	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 9222	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 9223	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 9224	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 9225	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 9226	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 9227	'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 9228	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 9229	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9230	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9231	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9232	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 9233	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 9234	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 9235	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 9236	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 9237	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 9238	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 7996.
Syntax	cxnn0 9239	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 9240	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 7996. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7119 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 9241	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 9242	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 9243	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 9244	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 9245	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 9246	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 9247	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 9248	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 9249	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 9250	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 9251	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 9252	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 9253	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 9254	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 9255	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 9256	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 9257	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 9258	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 9259	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR
Theorem	zssre 9260	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ RR
Theorem	zsscn 9261	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ CC
Theorem	zex 9262	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 9263	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 9264	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 9265	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 9266	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 9267	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 9268	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 9269	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 9270	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 9271	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 9272	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 9273	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 9274	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 9275	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ
Theorem	elnnz1 9276	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 9277	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN = { x e. ZZ | 1 <_ x }
Theorem	nn0zrab 9278	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN0 = { x e. ZZ | 0 <_ x }
Theorem	1z 9279	One is an integer. (Contributed by NM, 10-May-2004.)
|- 1 e. ZZ
Theorem	1zzd 9280	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. ZZ )
Theorem	2z 9281	Two is an integer. (Contributed by NM, 10-May-2004.)
|- 2 e. ZZ
Theorem	3z 9282	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. ZZ
Theorem	4z 9283	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|- 4 e. ZZ
Theorem	znegcl 9284	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> -u N e. ZZ )
Theorem	neg1z 9285	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 9286	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 9287	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 9288	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 9289	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 9290	Lemma for zaddcl 9293. 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 9291	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 9292	Lemma for zaddcl 9293. 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 9293	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 9294	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 9295	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 9296	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 9297	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 9298	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 9299	'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 9300	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 ) ) )