P. 94 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nn0supp 9301	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 9302	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9303	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9304	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9305	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 9306	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 9307	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 9308	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 9309	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 9310	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 9311	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 8065.
Syntax	cxnn0 9312	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 9313	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 8065. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7186 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 9314	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 9315	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 9316	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 9317	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 9318	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 9319	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 9320	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 9321	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 9322	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 9323	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 9324	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 9325	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 9326	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 9327	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 9328	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 9329	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 9330	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 9331	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 9332	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR
Theorem	zssre 9333	The integers are a subset of the reals. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ RR
Theorem	zsscn 9334	The integers are a subset of the complex numbers. (Contributed by NM, 2-Aug-2004.)
|- ZZ C_ CC
Theorem	zex 9335	The set of integers exists. (Contributed by NM, 30-Jul-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
|- ZZ e. _V
Theorem	elnnz 9336	Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)
|- ( N e. NN <-> ( N e. ZZ /\ 0 < N ) )
Theorem	0z 9337	Zero is an integer. (Contributed by NM, 12-Jan-2002.)
|- 0 e. ZZ
Theorem	0zd 9338	Zero is an integer, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 0 e. ZZ )
Theorem	elnn0z 9339	Nonnegative integer property expressed in terms of integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 <-> ( N e. ZZ /\ 0 <_ N ) )
Theorem	elznn0nn 9340	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 9341	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 9342	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 9343	Positive integers are a subset of integers. (Contributed by NM, 9-Jan-2002.)
|- NN C_ ZZ
Theorem	nn0ssz 9344	Nonnegative integers are a subset of the integers. (Contributed by NM, 9-May-2004.)
|- NN0 C_ ZZ
Theorem	nnz 9345	A positive integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN -> N e. ZZ )
Theorem	nn0z 9346	A nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> N e. ZZ )
Theorem	nnzi 9347	A positive integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN   =>    |- N e. ZZ
Theorem	nn0zi 9348	A nonnegative integer is an integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- N e. NN0   =>    |- N e. ZZ
Theorem	elnnz1 9349	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 9350	Positive integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN = { x e. ZZ | 1 <_ x }
Theorem	nn0zrab 9351	Nonnegative integers expressed as a subset of integers. (Contributed by NM, 3-Oct-2004.)
|- NN0 = { x e. ZZ | 0 <_ x }
Theorem	1z 9352	One is an integer. (Contributed by NM, 10-May-2004.)
|- 1 e. ZZ
Theorem	1zzd 9353	1 is an integer, deductive form (common case). (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( ph -> 1 e. ZZ )
Theorem	2z 9354	Two is an integer. (Contributed by NM, 10-May-2004.)
|- 2 e. ZZ
Theorem	3z 9355	3 is an integer. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. ZZ
Theorem	4z 9356	4 is an integer. (Contributed by BJ, 26-Mar-2020.)
|- 4 e. ZZ
Theorem	znegcl 9357	Closure law for negative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> -u N e. ZZ )
Theorem	neg1z 9358	-1 is an integer (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. ZZ
Theorem	znegclb 9359	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 9360	The negative of a nonnegative integer is an integer. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> -u N e. ZZ )
Theorem	nn0negzi 9361	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 9362	Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.)
|- ( N e. ZZ -> ( N + 1 ) e. ZZ )
Theorem	zaddcllempos 9363	Lemma for zaddcl 9366. 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 9364	"Reverse" second Peano postulate for integers. (Contributed by NM, 12-Sep-2005.)
|- ( N e. ZZ -> ( N - 1 ) e. ZZ )
Theorem	zaddcllemneg 9365	Lemma for zaddcl 9366. 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 9366	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 9367	Closure of subtraction of integers. (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
Theorem	ztri3or0 9368	Integer trichotomy (with zero). (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
Theorem	ztri3or 9369	Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )
Theorem	zletric 9370	Trichotomy law. (Contributed by Jim Kingdon, 27-Mar-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B <_ A ) )
Theorem	zlelttric 9371	Trichotomy law. (Contributed by Jim Kingdon, 17-Apr-2020.)
|- ( ( A e. ZZ /\ B e. ZZ ) -> ( A <_ B \/ B < A ) )
Theorem	zltnle 9372	'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 9373	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 9374	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 9375	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 9376	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 9377	The positive difference of unequal integers is a positive integer. (Generalization of nnsub 9029.) (Contributed by NM, 11-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
Theorem	nzadd 9378	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 9379	Closure of multiplication of integers. (Contributed by NM, 30-Jul-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
Theorem	zltp1le 9380	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 9381	Integer ordering relation. (Contributed by NM, 10-May-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	zlem1lt 9382	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( M - 1 ) < N ) )
Theorem	zltlem1 9383	Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	zgt0ge1 9384	An integer greater than 0 is greater than or equal to 1. (Contributed by AV, 14-Oct-2018.)
|- ( Z e. ZZ -> ( 0 < Z <-> 1 <_ Z ) )
Theorem	nnleltp1 9385	Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( A <_ B <-> A < ( B + 1 ) ) )
Theorem	nnltp1le 9386	Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
|- ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( A + 1 ) <_ B ) )
Theorem	nnaddm1cl 9387	Closure of addition of positive integers minus one. (Contributed by NM, 6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( A e. NN /\ B e. NN ) -> ( ( A + B ) - 1 ) e. NN )
Theorem	nn0ltp1le 9388	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> ( M + 1 ) <_ N ) )
Theorem	nn0leltp1 9389	Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> M < ( N + 1 ) ) )
Theorem	nn0ltlem1 9390	Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M < N <-> M <_ ( N - 1 ) ) )
Theorem	znn0sub 9391	The nonnegative difference of integers is a nonnegative integer. (Generalization of nn0sub 9392.) (Contributed by NM, 14-Jul-2005.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	nn0sub 9392	Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( ( M e. NN0 /\ N e. NN0 ) -> ( M <_ N <-> ( N - M ) e. NN0 ) )
Theorem	ltsubnn0 9393	Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( B < A -> ( A - B ) e. NN0 ) )
Theorem	nn0negleid 9394	A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021.)
|- ( A e. NN0 -> -u A <_ A )
Theorem	difgtsumgt 9395	If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
|- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) -> C < ( A + B ) ) )
Theorem	nn0n0n1ge2 9396	A nonnegative integer which is neither 0 nor 1 is greater than or equal to 2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
|- ( ( N e. NN0 /\ N =/= 0 /\ N =/= 1 ) -> 2 <_ N )
Theorem	elz2 9397*	Membership in the set of integers. Commonly used in constructions of the integers as equivalence classes under subtraction of the positive integers. (Contributed by Mario Carneiro, 16-May-2014.)
|- ( N e. ZZ <-> E. x e. NN E. y e. NN N = ( x - y ) )
Theorem	dfz2 9398	Alternate definition of the integers, based on elz2 9397. (Contributed by Mario Carneiro, 16-May-2014.)
|- ZZ = ( - " ( NN X. NN ) )
Theorem	nn0sub2 9399	Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
|- ( ( M e. NN0 /\ N e. NN0 /\ M <_ N ) -> ( N - M ) e. NN0 )
Theorem	zapne 9400	Apartness is equivalent to not equal for integers. (Contributed by Jim Kingdon, 14-Mar-2020.)
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M # N <-> M =/= N ) )