P. 95 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9401-9500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	4nn 9401	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9402	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9403	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9404	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9405	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9406	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9407	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9408	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9409	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9410	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9411	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9412	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9413	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9414	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9415	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9416	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9417	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9418	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9419	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6
Theorem	2lt6 9420	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 6
Theorem	1lt6 9421	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|- 1 < 6
Theorem	6lt7 9422	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 7
Theorem	5lt7 9423	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 7
Theorem	4lt7 9424	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 7
Theorem	3lt7 9425	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 7
Theorem	2lt7 9426	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 7
Theorem	1lt7 9427	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 7
Theorem	7lt8 9428	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 < 8
Theorem	6lt8 9429	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 8
Theorem	5lt8 9430	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 8
Theorem	4lt8 9431	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9432	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9433	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8
Theorem	1lt8 9434	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 8
Theorem	8lt9 9435	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- 8 < 9
Theorem	7lt9 9436	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 7 < 9
Theorem	6lt9 9437	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 6 < 9
Theorem	5lt9 9438	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 5 < 9
Theorem	4lt9 9439	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 4 < 9
Theorem	3lt9 9440	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 3 < 9
Theorem	2lt9 9441	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 2 < 9
Theorem	1lt9 9442	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
|- 1 < 9
Theorem	0ne2 9443	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 2
Theorem	1ne2 9444	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
|- 1 =/= 2
Theorem	1ap2 9445	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
Theorem	1le2 9446	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 2
Theorem	2cnne0 9447	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 2 e. CC /\ 2 =/= 0 )
Theorem	2rene0 9448	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 e. RR /\ 2 =/= 0 )
Theorem	1le3 9449	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 3
Theorem	neg1mulneg1e1 9450	-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 9451	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. RR
Theorem	halfcn 9452	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. CC
Theorem	halfgt0 9453	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
|- 0 < ( 1 / 2 )
Theorem	halfge0 9454	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
|- 0 <_ ( 1 / 2 )
Theorem	halflt1 9455	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
|- ( 1 / 2 ) < 1
Theorem	1mhlfehlf 9456	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
Theorem	8th4div3 9457	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )
Theorem	halfpm6th 9458	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 9459	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( _i x. 0 ) = 0
Theorem	2mulicn 9460	( 2 x. _i ) e. CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) e. CC
Theorem	iap0 9461	The imaginary unit _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- _i # 0
Theorem	2muliap0 9462	2 x. _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- ( 2 x. _i ) # 0
Theorem	2muline0 9463	( 2 x. _i ) =/= 0. See also 2muliap0 9462. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) =/= 0
4.4.5  Simple number properties
Theorem	halfcl 9464	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
|- ( A e. CC -> ( A / 2 ) e. CC )
Theorem	rehalfcl 9465	Real closure of half. (Contributed by NM, 1-Jan-2006.)
|- ( A e. RR -> ( A / 2 ) e. RR )
Theorem	half0 9466	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 9467	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
|- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	halfpos2 9468	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 9469	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 9470	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 9471	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 9472	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 9473	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 9474	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 9475	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 9476*	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 9477	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 9478	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 9479	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 9480	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 9481	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 2 x. A ) = ( A + A ) )
Theorem	times2d 9482	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 2 ) = ( A + A ) )
Theorem	halfcld 9483	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 9484	Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	rehalfcld 9485	Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( A / 2 ) e. RR )
Theorem	lt2halvesd 9486	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 9487	Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
|- A e. RR   =>    |- ( A / 2 ) e. RR
Theorem	add1p1 9488	Adding two times 1 to a number. (Contributed by AV, 22-Sep-2018.)
|- ( N e. CC -> ( ( N + 1 ) + 1 ) = ( N + 2 ) )
Theorem	sub1m1 9489	Subtracting two times 1 from a number. (Contributed by AV, 23-Oct-2018.)
|- ( N e. CC -> ( ( N - 1 ) - 1 ) = ( N - 2 ) )
Theorem	cnm2m1cnm3 9490	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 9491	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 9492	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 9493*	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 9494*	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 9495*	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 9496	Extend class notation to include the class of nonnegative integers.
class NN0
Definition	df-n0 9497	Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 = ( NN u. { 0 } )
Theorem	elnn0 9498	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 9499	Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN C_ NN0
Theorem	nn0ssre 9500	Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
|- NN0 C_ RR