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	4pos 9401	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9402	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9403	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9404	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9405	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9406	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9407	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9408	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 9409	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9410	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9411	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9412	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9413	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9414	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 9415	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 9416	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 9417	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 9418	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 9419	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 9420	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 9421	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2
Theorem	2m1e1 9422	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9450. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 2 - 1 ) = 1
Theorem	1e2m1 9423	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9424	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9425	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9426	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9427	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9428	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9429	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9430	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9431	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 9432	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 9433	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9434	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9435	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2txmxeqx 9436	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 9437	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9438	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9439	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9440	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9441	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9442	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9443	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9444	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9445	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9446	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9447	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9448	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9449	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9450	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9451	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9452	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9453	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9454	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9455	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9456	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9457	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9458	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9459	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9460	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9461	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9462	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9463	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9464	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9465	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9466	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9467	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9468	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9469	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN
Theorem	6nn 9470	6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 e. NN
Theorem	7nn 9471	7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 e. NN
Theorem	8nn 9472	8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 8 e. NN
Theorem	9nn 9473	9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
|- 9 e. NN
Theorem	1lt2 9474	1 is less than 2. (Contributed by NM, 24-Feb-2005.)
|- 1 < 2
Theorem	2lt3 9475	2 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 2 < 3
Theorem	1lt3 9476	1 is less than 3. (Contributed by NM, 26-Sep-2010.)
|- 1 < 3
Theorem	3lt4 9477	3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 4
Theorem	2lt4 9478	2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 4
Theorem	1lt4 9479	1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 4
Theorem	4lt5 9480	4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 5
Theorem	3lt5 9481	3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 5
Theorem	2lt5 9482	2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 5
Theorem	1lt5 9483	1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 5
Theorem	5lt6 9484	5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 6
Theorem	4lt6 9485	4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 6
Theorem	3lt6 9486	3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 6
Theorem	2lt6 9487	2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 6
Theorem	1lt6 9488	1 is less than 6. (Contributed by NM, 19-Oct-2012.)
|- 1 < 6
Theorem	6lt7 9489	6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 7
Theorem	5lt7 9490	5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 7
Theorem	4lt7 9491	4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 7
Theorem	3lt7 9492	3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 7
Theorem	2lt7 9493	2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 7
Theorem	1lt7 9494	1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 1 < 7
Theorem	7lt8 9495	7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 7 < 8
Theorem	6lt8 9496	6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 6 < 8
Theorem	5lt8 9497	5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 < 8
Theorem	4lt8 9498	4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 4 < 8
Theorem	3lt8 9499	3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 3 < 8
Theorem	2lt8 9500	2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 2 < 8