P. 94 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 9301-9400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1p0e1 9301	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 9302	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2
Theorem	2m1e1 9303	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9331. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 2 - 1 ) = 1
Theorem	1e2m1 9304	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9305	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9306	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9307	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9308	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9309	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9310	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9311	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9312	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 9313	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 9314	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9315	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9316	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2txmxeqx 9317	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 9318	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9319	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9320	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9321	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9322	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9323	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9324	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9325	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9326	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9327	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9328	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9329	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9330	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9331	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9332	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9333	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9334	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9335	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9336	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9337	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9338	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9339	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9340	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9341	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9342	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9343	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9344	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9345	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9346	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9347	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9348	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9349	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9350	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9351	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9352	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9353	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9354	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9355	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9356	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9357	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9358	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9359	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9360	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9361	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9362	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9363	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9364	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9365	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9366	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9367	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6
Theorem	2lt6 9368	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 6
Theorem	1lt6 9369	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|- 1 < 6
Theorem	6lt7 9370	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 7
Theorem	5lt7 9371	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 7
Theorem	4lt7 9372	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 7
Theorem	3lt7 9373	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 7
Theorem	2lt7 9374	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 7
Theorem	1lt7 9375	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 7
Theorem	7lt8 9376	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 < 8
Theorem	6lt8 9377	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 8
Theorem	5lt8 9378	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 8
Theorem	4lt8 9379	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9380	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9381	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8
Theorem	1lt8 9382	1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 8
Theorem	8lt9 9383	8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
|- 8 < 9
Theorem	7lt9 9384	7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 7 < 9
Theorem	6lt9 9385	6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 6 < 9
Theorem	5lt9 9386	5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 5 < 9
Theorem	4lt9 9387	4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 4 < 9
Theorem	3lt9 9388	3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 3 < 9
Theorem	2lt9 9389	2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- 2 < 9
Theorem	1lt9 9390	1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
|- 1 < 9
Theorem	0ne2 9391	0 is not equal to 2. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 2
Theorem	1ne2 9392	1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
|- 1 =/= 2
Theorem	1ap2 9393	1 is apart from 2. (Contributed by Jim Kingdon, 29-Oct-2022.)
|- 1 # 2
Theorem	1le2 9394	1 is less than or equal to 2 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 2
Theorem	2cnne0 9395	2 is a nonzero complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
|- ( 2 e. CC /\ 2 =/= 0 )
Theorem	2rene0 9396	2 is a nonzero real number (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 e. RR /\ 2 =/= 0 )
Theorem	1le3 9397	1 is less than or equal to 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 <_ 3
Theorem	neg1mulneg1e1 9398	-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 9399	One-half is real. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. RR
Theorem	halfcn 9400	One-half is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 / 2 ) e. CC