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	4lt8 9201	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9202	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9203	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8
Theorem	1lt8 9204	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 8
Theorem	8lt9 9205	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- 8 < 9
Theorem	7lt9 9206	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 7 < 9
Theorem	6lt9 9207	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 6 < 9
Theorem	5lt9 9208	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 5 < 9
Theorem	4lt9 9209	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 4 < 9
Theorem	3lt9 9210	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 3 < 9
Theorem	2lt9 9211	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 2 < 9
Theorem	1lt9 9212	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
|- 1 < 9
Theorem	0ne2 9213	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 2
Theorem	1ne2 9214	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
|- 1 =/= 2
Theorem	1ap2 9215	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
Theorem	1le2 9216	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 2
Theorem	2cnne0 9217	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 2 e. CC /\ 2 =/= 0 )
Theorem	2rene0 9218	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 e. RR /\ 2 =/= 0 )
Theorem	1le3 9219	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 3
Theorem	neg1mulneg1e1 9220	-u 1 x. -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( -u 1 x. -u 1 ) = 1
Theorem	halfre 9221	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. RR
Theorem	halfcn 9222	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. CC
Theorem	halfgt0 9223	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
|- 0 < ( 1 / 2 )
Theorem	halfge0 9224	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
|- 0 <_ ( 1 / 2 )
Theorem	halflt1 9225	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
|- ( 1 / 2 ) < 1
Theorem	1mhlfehlf 9226	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
Theorem	8th4div3 9227	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )
Theorem	halfpm6th 9228	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 9229	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( _i x. 0 ) = 0
Theorem	2mulicn 9230	( 2 x. _i ) e. CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) e. CC
Theorem	iap0 9231	The imaginary unit _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- _i # 0
Theorem	2muliap0 9232	2 x. _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- ( 2 x. _i ) # 0
Theorem	2muline0 9233	( 2 x. _i ) =/= 0. See also 2muliap0 9232. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) =/= 0
4.4.5  Simple number properties
Theorem	halfcl 9234	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
|- ( A e. CC -> ( A / 2 ) e. CC )
Theorem	rehalfcl 9235	Real closure of half. (Contributed by NM, 1-Jan-2006.)
|- ( A e. RR -> ( A / 2 ) e. RR )
Theorem	half0 9236	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 9237	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
|- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	halfpos2 9238	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 9239	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 9240	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 9241	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 9242	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 9243	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 9244	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 9245	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 9246*	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 9247	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 9248	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 9249	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 9250	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 9251	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 2 x. A ) = ( A + A ) )
Theorem	times2d 9252	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 2 ) = ( A + A ) )
Theorem	halfcld 9253	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 9254	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	rehalfcld 9255	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A / 2 ) e. RR )
Theorem	lt2halvesd 9256	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 9257	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- ( A / 2 ) e. RR
Theorem	add1p1 9258	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
|- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
Theorem	sub1m1 9259	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
|- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) )
Theorem	cnm2m1cnm3 9260	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 9261	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 9262	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 9263*	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 9264*	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 9265*	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 9266	Extend class notation to include the class of nonnegative integers.
class NN0
Definition	df-n0 9267	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 = ( NN u. { 0 } )
Theorem	elnn0 9268	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 9269	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN C_ NN0
Theorem	nn0ssre 9270	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 C_ RR
Theorem	nn0sscn 9271	Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
|- NN0 C_ CC
Theorem	nn0ex 9272	The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
|- NN0 e. _V
Theorem	nnnn0 9273	A positive integer is a nonnegative integer. (Contributed by NM, 9-May-2004.)
|- ( A e. NN -> A e. NN0 )
Theorem	nnnn0i 9274	A positive integer is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
|- N e. NN   =>    |- N e. NN0
Theorem	nn0re 9275	A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. RR )
Theorem	nn0cn 9276	A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
|- ( A e. NN0 -> A e. CC )
Theorem	nn0rei 9277	A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. RR
Theorem	nn0cni 9278	A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
|- A e. NN0   =>    |- A e. CC
Theorem	dfn2 9279	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 9280	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 9281	0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 0 e. NN0
Theorem	1nn0 9282	1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 1 e. NN0
Theorem	2nn0 9283	2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
|- 2 e. NN0
Theorem	3nn0 9284	3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 3 e. NN0
Theorem	4nn0 9285	4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
|- 4 e. NN0
Theorem	5nn0 9286	5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 5 e. NN0
Theorem	6nn0 9287	6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 6 e. NN0
Theorem	7nn0 9288	7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 7 e. NN0
Theorem	8nn0 9289	8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 8 e. NN0
Theorem	9nn0 9290	9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
|- 9 e. NN0
Theorem	nn0ge0 9291	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 9292	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 9293	Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
|- N e. NN0   =>    |- 0 <_ N
Theorem	nn0le0eq0 9294	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 9295	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 9296	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 9297	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 9298	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 9299	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 9300	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 )