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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegneg1e1 8001 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 8002 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 8003 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 8004 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 8005 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theorem1p0e1 8006 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 8007 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 8008 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 8030. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 8009 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 8010 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.)
(3 − 1) = 2

Theorem2p2e4 8011 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: http://us.metamath.org/mpeuni/mmset.html#trivia. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4

Theorem2times 8012 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 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))

Theoremtimes2 8013 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 8014 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 8015 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

Theorem2div2e1 8016 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

Theorem2p1e3 8017 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3

Theorem1p2e3 8018 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3

Theorem3p1e4 8019 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4

Theorem4p1e5 8020 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5

Theorem5p1e6 8021 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6

Theorem6p1e7 8022 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7

Theorem7p1e8 8023 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8

Theorem8p1e9 8024 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9

Theorem9p1e10 8025 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.)
(9 + 1) = 10

Theorem3p2e5 8026 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5

Theorem3p3e6 8027 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6

Theorem4p2e6 8028 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6

Theorem4p3e7 8029 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7

Theorem4p4e8 8030 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8

Theorem5p2e7 8031 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7

Theorem5p3e8 8032 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8

Theorem5p4e9 8033 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem5p5e10 8034 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.)
(5 + 5) = 10

Theorem6p2e8 8035 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 8036 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem6p4e10 8037 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.)
(6 + 4) = 10

Theorem7p2e9 8038 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem7p3e10 8039 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.)
(7 + 3) = 10

Theorem8p2e10 8040 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.)
(8 + 2) = 10

Theorem1t1e1 8041 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 8042 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

Theorem2t2e4 8043 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4

Theorem3t1e3 8044 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3

Theorem3t2e6 8045 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6

Theorem3t3e9 8046 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9

Theorem4t2e8 8047 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8

Theorem5t2e10 8048 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.)
(5 · 2) = 10

Theorem2t0e0 8049 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0

Theorem4d2e2 8050 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2

Theorem2nn 8051 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem3nn 8052 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem4nn 8053 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem5nn 8054 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem6nn 8055 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem7nn 8056 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem8nn 8057 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem9nn 8058 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem10nn 8059 10 is a positive integer. (Contributed by NM, 8-Nov-2012.)
10 ∈ ℕ

Theorem1lt2 8060 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2

Theorem2lt3 8061 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3

Theorem1lt3 8062 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3

Theorem3lt4 8063 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4

Theorem2lt4 8064 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4

Theorem1lt4 8065 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 8066 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 8067 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 8068 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 8069 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 8070 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 8071 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 8072 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 8073 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 8074 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 8075 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 8076 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 8077 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 8078 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 8079 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

Theorem1lt7 8080 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7

Theorem7lt8 8081 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8

Theorem6lt8 8082 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8

Theorem5lt8 8083 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8

Theorem4lt8 8084 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8

Theorem3lt8 8085 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8

Theorem2lt8 8086 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8

Theorem1lt8 8087 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8

Theorem8lt9 8088 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9

Theorem7lt9 8089 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9

Theorem6lt9 8090 6 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
6 < 9

Theorem5lt9 8091 5 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
5 < 9

Theorem4lt9 8092 4 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
4 < 9

Theorem3lt9 8093 3 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
3 < 9

Theorem2lt9 8094 2 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
2 < 9

Theorem1lt9 8095 1 is less than 9. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
1 < 9

Theorem9lt10 8096 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
9 < 10

Theorem8lt10 8097 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
8 < 10

Theorem7lt10 8098 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
7 < 10

Theorem6lt10 8099 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
6 < 10

Theorem5lt10 8100 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
5 < 10

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