P. 91 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9001-9100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	6cn 9001	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 9002	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9003	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9004	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9005	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9006	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9007	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9008	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9009	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9010	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9011	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9012	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9013	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 9014	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 9015	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 9016	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9017	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9018	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9019	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9020	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9021	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9022	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9023	The number 9 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 9
4.4.4  Some properties of specific numbers This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.
Theorem	neg1cn 9024	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9025	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9026	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9027	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9028	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9029	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 9030	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 9031	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 9032	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 9033	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 9034	Function value at N + 1 with N replaced by 0. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.)
|- ( N = 0 -> ( F ` ( N + 1 ) ) = ( F ` 1 ) )
Theorem	1p0e1 9035	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 9036	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2
Theorem	2m1e1 9037	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9064. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 2 - 1 ) = 1
Theorem	1e2m1 9038	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9039	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9040	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9041	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9042	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9043	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9044	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9045	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9046	Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: https://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
|- ( 2 + 2 ) = 4
Theorem	2times 9047	Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
|- ( A e. CC -> ( 2 x. A ) = ( A + A ) )
Theorem	times2 9048	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9049	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9050	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2div2e1 9051	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9052	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9053	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9054	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9055	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9056	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9057	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9058	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9059	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9060	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9061	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9062	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9063	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9064	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9065	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9066	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9067	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9068	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9069	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9070	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9071	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9072	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9073	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9074	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9075	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9076	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9077	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9078	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9079	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9080	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9081	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9082	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9083	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9084	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9085	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9086	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9087	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9088	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9089	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9090	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9091	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9092	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9093	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9094	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9095	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9096	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9097	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9098	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9099	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9100	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6