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	1e2m1 9101	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9102	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9103	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9104	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9105	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9106	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9107	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9108	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9109	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 9110	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 9111	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9112	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9113	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2div2e1 9114	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9115	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9116	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9117	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9118	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9119	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9120	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9121	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9122	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9123	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9124	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9125	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9126	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9127	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9128	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9129	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9130	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9131	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9132	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9133	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9134	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9135	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9136	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9137	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9138	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9139	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9140	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9141	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9142	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9143	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9144	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9145	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9146	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9147	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9148	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9149	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9150	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9151	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9152	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9153	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9154	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9155	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9156	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9157	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9158	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9159	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9160	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9161	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9162	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9163	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6
Theorem	2lt6 9164	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 6
Theorem	1lt6 9165	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|- 1 < 6
Theorem	6lt7 9166	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 7
Theorem	5lt7 9167	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 7
Theorem	4lt7 9168	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 7
Theorem	3lt7 9169	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 7
Theorem	2lt7 9170	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 7
Theorem	1lt7 9171	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 7
Theorem	7lt8 9172	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 < 8
Theorem	6lt8 9173	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 8
Theorem	5lt8 9174	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 8
Theorem	4lt8 9175	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9176	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9177	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8
Theorem	1lt8 9178	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 8
Theorem	8lt9 9179	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- 8 < 9
Theorem	7lt9 9180	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 7 < 9
Theorem	6lt9 9181	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 6 < 9
Theorem	5lt9 9182	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 5 < 9
Theorem	4lt9 9183	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 4 < 9
Theorem	3lt9 9184	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 3 < 9
Theorem	2lt9 9185	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 2 < 9
Theorem	1lt9 9186	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
|- 1 < 9
Theorem	0ne2 9187	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 2
Theorem	1ne2 9188	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
|- 1 =/= 2
Theorem	1ap2 9189	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
Theorem	1le2 9190	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 2
Theorem	2cnne0 9191	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 2 e. CC /\ 2 =/= 0 )
Theorem	2rene0 9192	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 e. RR /\ 2 =/= 0 )
Theorem	1le3 9193	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 3
Theorem	neg1mulneg1e1 9194	-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 9195	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. RR
Theorem	halfcn 9196	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. CC
Theorem	halfgt0 9197	One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
|- 0 < ( 1 / 2 )
Theorem	halfge0 9198	One-half is not negative. (Contributed by AV, 7-Jun-2020.)
|- 0 <_ ( 1 / 2 )
Theorem	halflt1 9199	One-half is less than one. (Contributed by NM, 24-Feb-2005.)
|- ( 1 / 2 ) < 1
Theorem	1mhlfehlf 9200	Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )