P. 96 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9501-9600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2halvesd 9501	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	rehalfcld 9502	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A / 2 ) e. RR )
Theorem	lt2halvesd 9503	A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   &    |- ( ph -> A < ( C / 2 ) )   &    |- ( ph -> B < ( C / 2 ) )   =>    |- ( ph -> ( A + B ) < C )
Theorem	rehalfcli 9504	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- ( A / 2 ) e. RR
Theorem	add1p1 9505	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
|- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
Theorem	sub1m1 9506	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
|- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) )
Theorem	cnm2m1cnm3 9507	Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
|- ( A e. CC -> ( ( A - 2 ) - 1 ) = ( A - 3 ) )
Theorem	xp1d2m1eqxm1d2 9508	A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020.)
|- ( X e. CC -> ( ( ( X + 1 ) / 2 ) - 1 ) = ( ( X - 1 ) / 2 ) )
Theorem	div4p1lem1div2 9509	An integer greater than 5, divided by 4 and increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)
|- ( ( N e. RR /\ 6 <_ N ) -> ( ( N / 4 ) + 1 ) <_ ( ( N - 1 ) / 2 ) )
4.4.6  The Archimedean property
Theorem	arch 9510*	Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
|- ( A e. RR -> E. n e. NN A < n )
Theorem	nnrecl 9511*	There exists a positive integer whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
|- ( ( A e. RR /\ 0 < A ) -> E. n e. NN ( 1 / n ) < A )
Theorem	bndndx 9512*	A bounded real sequence A ( k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
|- ( E. x e. RR A. k e. NN ( A e. RR /\ A <_ x ) -> E. k e. NN A <_ k )
4.4.7  Nonnegative integers (as a subset of complex numbers)
Syntax	cn0 9513	Extend class notation to include the class of nonnegative integers.
class NN0
Definition	df-n0 9514	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 = ( NN u. { 0 } )
Theorem	elnn0 9515	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 9516	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN C_ NN0
Theorem	nn0ssre 9517	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 C_ RR
Theorem	nn0sscn 9518	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
|- NN0 C_ CC
Theorem	nn0ex 9519	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
|- NN0 e. _V
Theorem	nnnn0 9520	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
|- ( A e. NN -> A e. NN0 )
Theorem	nnnn0i 9521	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
|- N e. NN   =>    |- N e. NN0
Theorem	nn0re 9522	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. RR )
Theorem	nn0cn 9523	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. CC )
Theorem	nn0rei 9524	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. RR
Theorem	nn0cni 9525	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. CC
Theorem	dfn2 9526	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 9527	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 9528	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 0 e. NN0
Theorem	1nn0 9529	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 1 e. NN0
Theorem	2nn0 9530	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 2 e. NN0
Theorem	3nn0 9531	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 3 e. NN0
Theorem	4nn0 9532	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 4 e. NN0
Theorem	5nn0 9533	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 5 e. NN0
Theorem	6nn0 9534	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 6 e. NN0
Theorem	7nn0 9535	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 7 e. NN0
Theorem	8nn0 9536	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 8 e. NN0
Theorem	9nn0 9537	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 9 e. NN0
Theorem	nn0ge0 9538	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 9539	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 9540	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- 0 <_ N
Theorem	nn0le0eq0 9541	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 9542	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 9543	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 9544	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 9545	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 9546	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 9547	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 9548	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 9549	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 9550	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 9551	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 9552	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 9553	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 9554	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 9555	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 9556	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 9557	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 9558	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 9559	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 9560	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 9561	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 9562	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 9563	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 9564	'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	fcdmnn0supp 9565	Two ways to write the support of a function into NN0. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by AV, 7-Jul-2019.)
|- ( ( I e. V /\ F : I --> NN0 ) -> ( F supp 0 ) = ( `' F " NN ) )
Theorem	fcdmnn0fsupp 9566	A function into NN0 is finitely supported iff its support is finite. (Contributed by AV, 8-Jul-2019.)
|- ( ( I e. V /\ F : I --> NN0 ) -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) )
Theorem	fcdmnn0suppg 9567	Version of fcdmnn0supp 9565 avoiding ax-coll 4230 by assuming F is a set rather than its domain I. (Contributed by SN, 5-Aug-2024.)
|- ( ( F e. V /\ F : I --> NN0 ) -> ( F supp 0 ) = ( `' F " NN ) )
Theorem	fcdmnn0fsuppg 9568	Version of fcdmnn0fsupp 9566 avoiding ax-coll 4230 by assuming F is a set rather than its domain I. (Contributed by SN, 5-Aug-2024.)
|- ( ( F e. V /\ F : I --> NN0 ) -> ( F finSupp 0 <-> ( `' F " NN ) e. Fin ) )
Theorem	nn0supp 9569	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 9570	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9571	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9572	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9573	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 9574	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 9575	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 9576	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 9577	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 9578	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 9579	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 8328.
Syntax	cxnn0 9580	The set of extended nonnegative integers.
class NN0*
Definition	df-xnn0 9581	Define the set of extended nonnegative integers that includes positive infinity. Analogue of the extension of the real numbers RR*, see df-xr 8328. If we assumed excluded middle, this would be essentially the same as ℕ∞ as defined at df-nninf 7424 but in its absence the relationship between the two is more complicated. (Contributed by AV, 10-Dec-2020.)
|- NN0* = ( NN0 u. { +oo } )
Theorem	elxnn0 9582	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 9583	The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.)
|- NN0 C_ NN0*
Theorem	nn0xnn0 9584	A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A e. NN0* )
Theorem	xnn0xr 9585	An extended nonnegative integer is an extended real. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A e. RR* )
Theorem	0xnn0 9586	Zero is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- 0 e. NN0*
Theorem	pnf0xnn0 9587	Positive infinity is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.)
|- +oo e. NN0*
Theorem	nn0nepnf 9588	No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0 -> A =/= +oo )
Theorem	nn0xnn0d 9589	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 9590	No standard nonnegative integer equals positive infinity, deduction form. (Contributed by AV, 10-Dec-2020.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A =/= +oo )
Theorem	xnn0nemnf 9591	No extended nonnegative integer equals negative infinity. (Contributed by AV, 10-Dec-2020.)
|- ( A e. NN0* -> A =/= -oo )
Theorem	xnn0xrnemnf 9592	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 9593	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 9594	Extend class notation to include the class of integers.
class ZZ
Definition	df-z 9595	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 9596	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 9597	The negative of a positive integer is an integer. (Contributed by NM, 12-Jan-2002.)
|- ( N e. NN -> -u N e. ZZ )
Theorem	zre 9598	An integer is a real. (Contributed by NM, 8-Jan-2002.)
|- ( N e. ZZ -> N e. RR )
Theorem	zcn 9599	An integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( N e. ZZ -> N e. CC )
Theorem	zrei 9600	An integer is a real number. (Contributed by NM, 14-Jul-2005.)
|- A e. ZZ   =>    |- A e. RR