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
Definition	df-9 9101	Define the number 9. (Contributed by NM, 27-May-1999.)
|- 9 = ( 8 + 1 )
Theorem	0ne1 9102	0 =/= 1 (common case). See aso 1ap0 8662. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 0 =/= 1
Theorem	1ne0 9103	1 =/= 0. See aso 1ap0 8662. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 1 =/= 0
Theorem	1m1e0 9104	( 1 - 1 ) = 0 (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 - 1 ) = 0
Theorem	2re 9105	The number 2 is real. (Contributed by NM, 27-May-1999.)
|- 2 e. RR
Theorem	2cn 9106	The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.)
|- 2 e. CC
Theorem	2ex 9107	2 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 2 e. _V
Theorem	2cnd 9108	2 is a complex number, deductive form (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( ph -> 2 e. CC )
Theorem	3re 9109	The number 3 is real. (Contributed by NM, 27-May-1999.)
|- 3 e. RR
Theorem	3cn 9110	The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.)
|- 3 e. CC
Theorem	3ex 9111	3 is a set (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 3 e. _V
Theorem	4re 9112	The number 4 is real. (Contributed by NM, 27-May-1999.)
|- 4 e. RR
Theorem	4cn 9113	The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 4 e. CC
Theorem	5re 9114	The number 5 is real. (Contributed by NM, 27-May-1999.)
|- 5 e. RR
Theorem	5cn 9115	The number 5 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 5 e. CC
Theorem	6re 9116	The number 6 is real. (Contributed by NM, 27-May-1999.)
|- 6 e. RR
Theorem	6cn 9117	The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 6 e. CC
Theorem	7re 9118	The number 7 is real. (Contributed by NM, 27-May-1999.)
|- 7 e. RR
Theorem	7cn 9119	The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 7 e. CC
Theorem	8re 9120	The number 8 is real. (Contributed by NM, 27-May-1999.)
|- 8 e. RR
Theorem	8cn 9121	The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 8 e. CC
Theorem	9re 9122	The number 9 is real. (Contributed by NM, 27-May-1999.)
|- 9 e. RR
Theorem	9cn 9123	The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 9 e. CC
Theorem	0le0 9124	Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- 0 <_ 0
Theorem	0le2 9125	0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
|- 0 <_ 2
Theorem	2pos 9126	The number 2 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 2
Theorem	2ne0 9127	The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
|- 2 =/= 0
Theorem	2ap0 9128	The number 2 is apart from zero. (Contributed by Jim Kingdon, 9-Mar-2020.)
|- 2 # 0
Theorem	3pos 9129	The number 3 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 3
Theorem	3ne0 9130	The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- 3 =/= 0
Theorem	3ap0 9131	The number 3 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 3 # 0
Theorem	4pos 9132	The number 4 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 4
Theorem	4ne0 9133	The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
|- 4 =/= 0
Theorem	4ap0 9134	The number 4 is apart from zero. (Contributed by Jim Kingdon, 10-Oct-2021.)
|- 4 # 0
Theorem	5pos 9135	The number 5 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 5
Theorem	6pos 9136	The number 6 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 6
Theorem	7pos 9137	The number 7 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 7
Theorem	8pos 9138	The number 8 is positive. (Contributed by NM, 27-May-1999.)
|- 0 < 8
Theorem	9pos 9139	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 9140	-1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
|- -u 1 e. CC
Theorem	neg1rr 9141	-1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
|- -u 1 e. RR
Theorem	neg1ne0 9142	-1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 =/= 0
Theorem	neg1lt0 9143	-1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u 1 < 0
Theorem	neg1ap0 9144	-1 is apart from zero. (Contributed by Jim Kingdon, 9-Jun-2020.)
|- -u 1 # 0
Theorem	negneg1e1 9145	-u -u 1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- -u -u 1 = 1
Theorem	1pneg1e0 9146	1 + -u 1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + -u 1 ) = 0
Theorem	0m0e0 9147	0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 0 - 0 ) = 0
Theorem	1m0e1 9148	1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 - 0 ) = 1
Theorem	0p1e1 9149	0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 0 + 1 ) = 1
Theorem	fv0p1e1 9150	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 9151	1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 0 ) = 1
Theorem	1p1e2 9152	1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
|- ( 1 + 1 ) = 2
Theorem	2m1e1 9153	2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9181. (Contributed by David A. Wheeler, 4-Jan-2017.)
|- ( 2 - 1 ) = 1
Theorem	1e2m1 9154	1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- 1 = ( 2 - 1 )
Theorem	3m1e2 9155	3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
|- ( 3 - 1 ) = 2
Theorem	4m1e3 9156	4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
|- ( 4 - 1 ) = 3
Theorem	5m1e4 9157	5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
|- ( 5 - 1 ) = 4
Theorem	6m1e5 9158	6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
|- ( 6 - 1 ) = 5
Theorem	7m1e6 9159	7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
|- ( 7 - 1 ) = 6
Theorem	8m1e7 9160	8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
|- ( 8 - 1 ) = 7
Theorem	9m1e8 9161	9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
|- ( 9 - 1 ) = 8
Theorem	2p2e4 9162	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 9163	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 9164	A number times 2. (Contributed by NM, 16-Oct-2007.)
|- ( A e. CC -> ( A x. 2 ) = ( A + A ) )
Theorem	2timesi 9165	Two times a number. (Contributed by NM, 1-Aug-1999.)
|- A e. CC   =>    |- ( 2 x. A ) = ( A + A )
Theorem	times2i 9166	A number times 2. (Contributed by NM, 11-May-2004.)
|- A e. CC   =>    |- ( A x. 2 ) = ( A + A )
Theorem	2txmxeqx 9167	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 9168	2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 / 2 ) = 1
Theorem	2p1e3 9169	2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 2 + 1 ) = 3
Theorem	1p2e3 9170	1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 1 + 2 ) = 3
Theorem	3p1e4 9171	3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 3 + 1 ) = 4
Theorem	4p1e5 9172	4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 4 + 1 ) = 5
Theorem	5p1e6 9173	5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 5 + 1 ) = 6
Theorem	6p1e7 9174	6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 6 + 1 ) = 7
Theorem	7p1e8 9175	7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 7 + 1 ) = 8
Theorem	8p1e9 9176	8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
|- ( 8 + 1 ) = 9
Theorem	3p2e5 9177	3 + 2 = 5. (Contributed by NM, 11-May-2004.)
|- ( 3 + 2 ) = 5
Theorem	3p3e6 9178	3 + 3 = 6. (Contributed by NM, 11-May-2004.)
|- ( 3 + 3 ) = 6
Theorem	4p2e6 9179	4 + 2 = 6. (Contributed by NM, 11-May-2004.)
|- ( 4 + 2 ) = 6
Theorem	4p3e7 9180	4 + 3 = 7. (Contributed by NM, 11-May-2004.)
|- ( 4 + 3 ) = 7
Theorem	4p4e8 9181	4 + 4 = 8. (Contributed by NM, 11-May-2004.)
|- ( 4 + 4 ) = 8
Theorem	5p2e7 9182	5 + 2 = 7. (Contributed by NM, 11-May-2004.)
|- ( 5 + 2 ) = 7
Theorem	5p3e8 9183	5 + 3 = 8. (Contributed by NM, 11-May-2004.)
|- ( 5 + 3 ) = 8
Theorem	5p4e9 9184	5 + 4 = 9. (Contributed by NM, 11-May-2004.)
|- ( 5 + 4 ) = 9
Theorem	6p2e8 9185	6 + 2 = 8. (Contributed by NM, 11-May-2004.)
|- ( 6 + 2 ) = 8
Theorem	6p3e9 9186	6 + 3 = 9. (Contributed by NM, 11-May-2004.)
|- ( 6 + 3 ) = 9
Theorem	7p2e9 9187	7 + 2 = 9. (Contributed by NM, 11-May-2004.)
|- ( 7 + 2 ) = 9
Theorem	1t1e1 9188	1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
|- ( 1 x. 1 ) = 1
Theorem	2t1e2 9189	2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
|- ( 2 x. 1 ) = 2
Theorem	2t2e4 9190	2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
|- ( 2 x. 2 ) = 4
Theorem	3t1e3 9191	3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 3 x. 1 ) = 3
Theorem	3t2e6 9192	3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
|- ( 3 x. 2 ) = 6
Theorem	3t3e9 9193	3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
|- ( 3 x. 3 ) = 9
Theorem	4t2e8 9194	4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
|- ( 4 x. 2 ) = 8
Theorem	2t0e0 9195	2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( 2 x. 0 ) = 0
Theorem	4d2e2 9196	One half of four is two. (Contributed by NM, 3-Sep-1999.)
|- ( 4 / 2 ) = 2
Theorem	2nn 9197	2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
|- 2 e. NN
Theorem	3nn 9198	3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 3 e. NN
Theorem	4nn 9199	4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
|- 4 e. NN
Theorem	5nn 9200	5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
|- 5 e. NN