Home | Metamath
Proof Explorer Theorem List (p. 112 of 426) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-27775) |
Hilbert Space Explorer
(27776-29300) |
Users' Mathboxes
(29301-42551) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 6re 11101 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
⊢ 6 ∈ ℝ | ||
Theorem | 6cn 11102 | The number 6 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 6 ∈ ℂ | ||
Theorem | 7re 11103 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
⊢ 7 ∈ ℝ | ||
Theorem | 7cn 11104 | The number 7 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 7 ∈ ℂ | ||
Theorem | 8re 11105 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
⊢ 8 ∈ ℝ | ||
Theorem | 8cn 11106 | The number 8 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 8 ∈ ℂ | ||
Theorem | 9re 11107 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
⊢ 9 ∈ ℝ | ||
Theorem | 9cn 11108 | The number 9 is complex. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 9 ∈ ℂ | ||
Theorem | 10reOLD 11109 | Obsolete version of 10re 11517 as of 8-Sep-2021. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 10 ∈ ℝ | ||
Theorem | 0le0 11110 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ≤ 0 | ||
Theorem | 0le2 11111 | 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ 0 ≤ 2 | ||
Theorem | 2pos 11112 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 2 | ||
Theorem | 2ne0 11113 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
⊢ 2 ≠ 0 | ||
Theorem | 3pos 11114 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 3 | ||
Theorem | 3ne0 11115 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ 3 ≠ 0 | ||
Theorem | 4pos 11116 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 4 | ||
Theorem | 4ne0 11117 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ 4 ≠ 0 | ||
Theorem | 5pos 11118 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 5 | ||
Theorem | 6pos 11119 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 6 | ||
Theorem | 7pos 11120 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 7 | ||
Theorem | 8pos 11121 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 8 | ||
Theorem | 9pos 11122 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 9 | ||
Theorem | 10posOLD 11123 | The number 10 is positive. (Contributed by NM, 5-Feb-2007.) Obsolete version of 10pos 11515 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 0 < 10 | ||
This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering). | ||
Theorem | neg1cn 11124 | -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ -1 ∈ ℂ | ||
Theorem | neg1rr 11125 | -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℝ | ||
Theorem | neg1ne0 11126 | -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 ≠ 0 | ||
Theorem | neg1lt0 11127 | -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 < 0 | ||
Theorem | negneg1e1 11128 | --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ --1 = 1 | ||
Theorem | 1pneg1e0 11129 | 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + -1) = 0 | ||
Theorem | 0m0e0 11130 | 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0 − 0) = 0 | ||
Theorem | 1m0e1 11131 | 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 − 0) = 1 | ||
Theorem | 0p1e1 11132 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (0 + 1) = 1 | ||
Theorem | 1p0e1 11133 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 0) = 1 | ||
Theorem | 1p1e2 11134 | 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) |
⊢ (1 + 1) = 2 | ||
Theorem | 2m1e1 11135 | 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 11164. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (2 − 1) = 1 | ||
Theorem | 1e2m1 11136 | 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 = (2 − 1) | ||
Theorem | 3m1e2 11137 | 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (3 − 1) = 2 | ||
Theorem | 4m1e3 11138 | 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (4 − 1) = 3 | ||
Theorem | 5m1e4 11139 | 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.) |
⊢ (5 − 1) = 4 | ||
Theorem | 6m1e5 11140 | 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.) |
⊢ (6 − 1) = 5 | ||
Theorem | 7m1e6 11141 | 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.) |
⊢ (7 − 1) = 6 | ||
Theorem | 8m1e7 11142 | 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (8 − 1) = 7 | ||
Theorem | 9m1e8 11143 | 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.) |
⊢ (9 − 1) = 8 | ||
Theorem | 2p2e4 11144 | Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 7621 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.) |
⊢ (2 + 2) = 4 | ||
Theorem | 2times 11145 | 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)) | ||
Theorem | times2 11146 | A number times 2. (Contributed by NM, 16-Oct-2007.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | 2timesi 11147 | Two times a number. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (2 · 𝐴) = (𝐴 + 𝐴) | ||
Theorem | times2i 11148 | A number times 2. (Contributed by NM, 11-May-2004.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 2) = (𝐴 + 𝐴) | ||
Theorem | 2txmxeqx 11149 | Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.) |
⊢ (𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋) | ||
Theorem | 2div2e1 11150 | 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 / 2) = 1 | ||
Theorem | 2p1e3 11151 | 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (2 + 1) = 3 | ||
Theorem | 1p2e3 11152 | 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 2) = 3 | ||
Theorem | 3p1e4 11153 | 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (3 + 1) = 4 | ||
Theorem | 4p1e5 11154 | 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (4 + 1) = 5 | ||
Theorem | 5p1e6 11155 | 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (5 + 1) = 6 | ||
Theorem | 6p1e7 11156 | 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (6 + 1) = 7 | ||
Theorem | 7p1e8 11157 | 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (7 + 1) = 8 | ||
Theorem | 8p1e9 11158 | 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (8 + 1) = 9 | ||
Theorem | 9p1e10OLD 11159 | 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) Obsolete version of 9p1e10 11496 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (9 + 1) = 10 | ||
Theorem | 3p2e5 11160 | 3 + 2 = 5. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 2) = 5 | ||
Theorem | 3p3e6 11161 | 3 + 3 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 3) = 6 | ||
Theorem | 4p2e6 11162 | 4 + 2 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 2) = 6 | ||
Theorem | 4p3e7 11163 | 4 + 3 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 3) = 7 | ||
Theorem | 4p4e8 11164 | 4 + 4 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 4) = 8 | ||
Theorem | 5p2e7 11165 | 5 + 2 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 2) = 7 | ||
Theorem | 5p3e8 11166 | 5 + 3 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 3) = 8 | ||
Theorem | 5p4e9 11167 | 5 + 4 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 4) = 9 | ||
Theorem | 5p5e10OLD 11168 | 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5p5e10 11596 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (5 + 5) = 10 | ||
Theorem | 6p2e8 11169 | 6 + 2 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 2) = 8 | ||
Theorem | 6p3e9 11170 | 6 + 3 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 3) = 9 | ||
Theorem | 6p4e10OLD 11171 | 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 6p4e10 11598 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (6 + 4) = 10 | ||
Theorem | 7p2e9 11172 | 7 + 2 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (7 + 2) = 9 | ||
Theorem | 7p3e10OLD 11173 | 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 7p3e10 11603 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (7 + 3) = 10 | ||
Theorem | 8p2e10OLD 11174 | 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 8p2e10 11610 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (8 + 2) = 10 | ||
Theorem | 1t1e1 11175 | 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 · 1) = 1 | ||
Theorem | 2t1e2 11176 | 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (2 · 1) = 2 | ||
Theorem | 2t2e4 11177 | 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) |
⊢ (2 · 2) = 4 | ||
Theorem | 3t1e3 11178 | 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (3 · 1) = 3 | ||
Theorem | 3t2e6 11179 | 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) |
⊢ (3 · 2) = 6 | ||
Theorem | 3t3e9 11180 | 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) |
⊢ (3 · 3) = 9 | ||
Theorem | 4t2e8 11181 | 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.) |
⊢ (4 · 2) = 8 | ||
Theorem | 5t2e10OLD 11182 | 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5t2e10 11634 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (5 · 2) = 10 | ||
Theorem | 2t0e0 11183 | 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · 0) = 0 | ||
Theorem | 4d2e2 11184 | One half of four is two. (Contributed by NM, 3-Sep-1999.) |
⊢ (4 / 2) = 2 | ||
Theorem | 2nn 11185 | 2 is a positive integer. (Contributed by NM, 20-Aug-2001.) |
⊢ 2 ∈ ℕ | ||
Theorem | 3nn 11186 | 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 3 ∈ ℕ | ||
Theorem | 4nn 11187 | 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 4 ∈ ℕ | ||
Theorem | 5nn 11188 | 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 ∈ ℕ | ||
Theorem | 6nn 11189 | 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 ∈ ℕ | ||
Theorem | 7nn 11190 | 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 ∈ ℕ | ||
Theorem | 8nn 11191 | 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 8 ∈ ℕ | ||
Theorem | 9nn 11192 | 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
⊢ 9 ∈ ℕ | ||
Theorem | 10nnOLD 11193 | Obsolete version of 10nn 11514 as of 6-Sep-2021. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 10 ∈ ℕ | ||
Theorem | 1lt2 11194 | 1 is less than 2. (Contributed by NM, 24-Feb-2005.) |
⊢ 1 < 2 | ||
Theorem | 2lt3 11195 | 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 2 < 3 | ||
Theorem | 1lt3 11196 | 1 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 1 < 3 | ||
Theorem | 3lt4 11197 | 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 4 | ||
Theorem | 2lt4 11198 | 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 4 | ||
Theorem | 1lt4 11199 | 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 1 < 4 | ||
Theorem | 4lt5 11200 | 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 4 < 5 |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |