P. 92 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	7re 9101	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9102	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9103	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9104	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9105	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9106	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9107	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9108	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9109	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9110	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9111	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9112	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 9113	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 9114	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 9115	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9116	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9117	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9118	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9119	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9120	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9121	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9122	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 9123	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9124	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9125	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9126	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9127	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9128	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 9129	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 9130	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 9131	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 9132	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 9133	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 9134	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 9135	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2
Theorem	2m1e1 9136	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9164. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 2 - 1 ) = 1
Theorem	1e2m1 9137	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9138	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9139	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9140	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9141	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9142	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9143	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9144	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9145	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 9146	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 9147	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9148	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9149	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2txmxeqx 9150	Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
|- ( X e. CC -> ( ( 2 x. X ) - X ) = X )
Theorem	2div2e1 9151	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9152	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9153	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9154	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9155	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9156	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9157	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9158	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9159	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9160	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9161	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9162	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9163	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9164	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9165	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9166	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9167	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9168	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9169	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9170	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9171	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9172	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9173	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9174	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9175	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9176	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9177	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9178	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9179	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9180	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9181	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9182	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9183	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9184	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9185	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9186	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9187	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9188	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9189	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9190	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9191	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9192	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9193	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9194	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9195	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9196	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9197	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9198	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9199	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9200	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6