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	8th4div3 9201	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )
Theorem	halfpm6th 9202	One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
|- ( ( ( 1 / 2 ) - ( 1 / 6 ) ) = ( 1 / 3 ) /\ ( ( 1 / 2 ) + ( 1 / 6 ) ) = ( 2 / 3 ) )
Theorem	it0e0 9203	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( _i x. 0 ) = 0
Theorem	2mulicn 9204	( 2 x. _i ) e. CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) e. CC
Theorem	iap0 9205	The imaginary unit _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- _i # 0
Theorem	2muliap0 9206	2 x. _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- ( 2 x. _i ) # 0
Theorem	2muline0 9207	( 2 x. _i ) =/= 0. See also 2muliap0 9206. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) =/= 0
4.4.5  Simple number properties
Theorem	halfcl 9208	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
|- ( A e. CC -> ( A / 2 ) e. CC )
Theorem	rehalfcl 9209	Real closure of half. (Contributed by NM, 1-Jan-2006.)
|- ( A e. RR -> ( A / 2 ) e. RR )
Theorem	half0 9210	Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
|- ( A e. CC -> ( ( A / 2 ) = 0 <-> A = 0 ) )
Theorem	2halves 9211	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
|- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	halfpos2 9212	A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
|- ( A e. RR -> ( 0 < A <-> 0 < ( A / 2 ) ) )
Theorem	halfpos 9213	A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
|- ( A e. RR -> ( 0 < A <-> ( A / 2 ) < A ) )
Theorem	halfnneg2 9214	A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
|- ( A e. RR -> ( 0 <_ A <-> 0 <_ ( A / 2 ) ) )
Theorem	halfaddsubcl 9215	Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) / 2 ) e. CC /\ ( ( A - B ) / 2 ) e. CC ) )
Theorem	halfaddsub 9216	Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A + B ) / 2 ) + ( ( A - B ) / 2 ) ) = A /\ ( ( ( A + B ) / 2 ) - ( ( A - B ) / 2 ) ) = B ) )
Theorem	subhalfhalf 9217	Subtracting the half of a number from the number yields the half of the number. (Contributed by AV, 28-Jun-2021.)
|- ( A e. CC -> ( A - ( A / 2 ) ) = ( A / 2 ) )
Theorem	lt2halves 9218	A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < ( C / 2 ) /\ B < ( C / 2 ) ) -> ( A + B ) < C ) )
Theorem	addltmul 9219	Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )
Theorem	nominpos 9220*	There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
|- -. E. x e. RR ( 0 < x /\ -. E. y e. RR ( 0 < y /\ y < x ) )
Theorem	avglt1 9221	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A < ( ( A + B ) / 2 ) ) )
Theorem	avglt2 9222	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
|- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( ( A + B ) / 2 ) < B ) )
Theorem	avgle1 9223	Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> A <_ ( ( A + B ) / 2 ) ) )
Theorem	avgle2 9224	Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( ( A + B ) / 2 ) <_ B ) )
Theorem	2timesd 9225	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 2 x. A ) = ( A + A ) )
Theorem	times2d 9226	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 2 ) = ( A + A ) )
Theorem	halfcld 9227	Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A / 2 ) e. CC )
Theorem	2halvesd 9228	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	rehalfcld 9229	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A / 2 ) e. RR )
Theorem	lt2halvesd 9230	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 9231	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- ( A / 2 ) e. RR
Theorem	add1p1 9232	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
|- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
Theorem	sub1m1 9233	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
|- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) )
Theorem	cnm2m1cnm3 9234	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 9235	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 9236	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 9237*	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 9238*	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 9239*	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 9240	Extend class notation to include the class of nonnegative integers.
class NN0
Definition	df-n0 9241	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 = ( NN u. { 0 } )
Theorem	elnn0 9242	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 9243	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN C_ NN0
Theorem	nn0ssre 9244	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 C_ RR
Theorem	nn0sscn 9245	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
|- NN0 C_ CC
Theorem	nn0ex 9246	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
|- NN0 e. _V
Theorem	nnnn0 9247	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
|- ( A e. NN -> A e. NN0 )
Theorem	nnnn0i 9248	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
|- N e. NN   =>    |- N e. NN0
Theorem	nn0re 9249	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. RR )
Theorem	nn0cn 9250	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. CC )
Theorem	nn0rei 9251	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. RR
Theorem	nn0cni 9252	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. CC
Theorem	dfn2 9253	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 9254	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 9255	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 0 e. NN0
Theorem	1nn0 9256	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 1 e. NN0
Theorem	2nn0 9257	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 2 e. NN0
Theorem	3nn0 9258	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 3 e. NN0
Theorem	4nn0 9259	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 4 e. NN0
Theorem	5nn0 9260	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 5 e. NN0
Theorem	6nn0 9261	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 6 e. NN0
Theorem	7nn0 9262	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 7 e. NN0
Theorem	8nn0 9263	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 8 e. NN0
Theorem	9nn0 9264	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 9 e. NN0
Theorem	nn0ge0 9265	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 9266	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 9267	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- 0 <_ N
Theorem	nn0le0eq0 9268	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 9269	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 9270	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 9271	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 9272	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 9273	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 9274	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 9275	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 9276	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 9277	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 9278	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 9279	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 9280	Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
|- ( N e. NN0 -> ( N + 1 ) e. NN0 )
Theorem	nnm1nn0 9281	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 9282	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 9283	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 9284	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 9285	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 9286	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 9287	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 9288	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 9289	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 9290	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 9291	'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 9292	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 9293	A positive integer is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN )   =>    |- ( ph -> A e. NN0 )
Theorem	nn0red 9294	A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. RR )
Theorem	nn0cnd 9295	A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. NN0 )   =>    |- ( ph -> A e. CC )
Theorem	nn0ge0d 9296	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 9297	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 9298	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 9299	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 9300	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 )