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	5p2e7 9401	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9402	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9403	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9404	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9405	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9406	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9407	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9408	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9409	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9410	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9411	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9412	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9413	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9414	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9415	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9416	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9417	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9418	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9419	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9420	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9421	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9422	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9423	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9424	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9425	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9426	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9427	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9428	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9429	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9430	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9431	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9432	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9433	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9434	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9435	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9436	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6
Theorem	2lt6 9437	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 6
Theorem	1lt6 9438	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|- 1 < 6
Theorem	6lt7 9439	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 7
Theorem	5lt7 9440	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 7
Theorem	4lt7 9441	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 7
Theorem	3lt7 9442	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 7
Theorem	2lt7 9443	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 7
Theorem	1lt7 9444	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 7
Theorem	7lt8 9445	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 < 8
Theorem	6lt8 9446	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 8
Theorem	5lt8 9447	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 8
Theorem	4lt8 9448	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9449	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9450	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8
Theorem	1lt8 9451	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 8
Theorem	8lt9 9452	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- 8 < 9
Theorem	7lt9 9453	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 7 < 9
Theorem	6lt9 9454	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 6 < 9
Theorem	5lt9 9455	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 5 < 9
Theorem	4lt9 9456	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 4 < 9
Theorem	3lt9 9457	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 3 < 9
Theorem	2lt9 9458	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 2 < 9
Theorem	1lt9 9459	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
|- 1 < 9
Theorem	0ne2 9460	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 2
Theorem	1ne2 9461	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
|- 1 =/= 2
Theorem	1ap2 9462	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
Theorem	1le2 9463	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 2
Theorem	2cnne0 9464	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 2 e. CC /\ 2 =/= 0 )
Theorem	2rene0 9465	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 e. RR /\ 2 =/= 0 )
Theorem	1le3 9466	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 3
Theorem	neg1mulneg1e1 9467	-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 9468	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. RR
Theorem	halfcn 9469	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. CC
Theorem	halfgt0 9470	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
|- 0 < ( 1 / 2 )
Theorem	halfge0 9471	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
|- 0 <_ ( 1 / 2 )
Theorem	halflt1 9472	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
|- ( 1 / 2 ) < 1
Theorem	1mhlfehlf 9473	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
Theorem	8th4div3 9474	An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
|- ( ( 1 / 8 ) x. ( 4 / 3 ) ) = ( 1 / 6 )
Theorem	halfpm6th 9475	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 9476	i times 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( _i x. 0 ) = 0
Theorem	2mulicn 9477	( 2 x. _i ) e. CC (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) e. CC
Theorem	iap0 9478	The imaginary unit _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- _i # 0
Theorem	2muliap0 9479	2 x. _i is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- ( 2 x. _i ) # 0
Theorem	2muline0 9480	( 2 x. _i ) =/= 0. See also 2muliap0 9479. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. _i ) =/= 0
4.4.5  Simple number properties
Theorem	halfcl 9481	Closure of half of a number (common case). (Contributed by NM, 1-Jan-2006.)
|- ( A e. CC -> ( A / 2 ) e. CC )
Theorem	rehalfcl 9482	Real closure of half. (Contributed by NM, 1-Jan-2006.)
|- ( A e. RR -> ( A / 2 ) e. RR )
Theorem	half0 9483	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 9484	Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
|- ( A e. CC -> ( ( A / 2 ) + ( A / 2 ) ) = A )
Theorem	halfpos2 9485	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 9486	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 9487	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 9488	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 9489	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 9490	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 9491	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 9492	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 9493*	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 9494	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 9495	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 9496	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 9497	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 9498	Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 2 x. A ) = ( A + A ) )
Theorem	times2d 9499	A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( A x. 2 ) = ( A + A ) )
Theorem	halfcld 9500	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 )