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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 2cnALT 11701 | Alternate proof of 2cn 11700. Shorter but uses more axioms. Similar proofs are possible for 3cn 11706, ... , 9cn 11725. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 2 ∈ ℂ | ||
Theorem | 2ex 11702 | The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 2 ∈ V | ||
Theorem | 2cnd 11703 | The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (𝜑 → 2 ∈ ℂ) | ||
Theorem | 3nn 11704 | 3 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 3 ∈ ℕ | ||
Theorem | 3re 11705 | The number 3 is real. (Contributed by NM, 27-May-1999.) |
⊢ 3 ∈ ℝ | ||
Theorem | 3cn 11706 | The number 3 is a complex number. (Contributed by FL, 17-Oct-2010.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 3 ∈ ℂ | ||
Theorem | 3ex 11707 | The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 3 ∈ V | ||
Theorem | 4nn 11708 | 4 is a positive integer. (Contributed by NM, 8-Jan-2006.) |
⊢ 4 ∈ ℕ | ||
Theorem | 4re 11709 | The number 4 is real. (Contributed by NM, 27-May-1999.) |
⊢ 4 ∈ ℝ | ||
Theorem | 4cn 11710 | The number 4 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 4 ∈ ℂ | ||
Theorem | 5nn 11711 | 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 5 ∈ ℕ | ||
Theorem | 5re 11712 | The number 5 is real. (Contributed by NM, 27-May-1999.) |
⊢ 5 ∈ ℝ | ||
Theorem | 5cn 11713 | The number 5 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 5 ∈ ℂ | ||
Theorem | 6nn 11714 | 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 6 ∈ ℕ | ||
Theorem | 6re 11715 | The number 6 is real. (Contributed by NM, 27-May-1999.) |
⊢ 6 ∈ ℝ | ||
Theorem | 6cn 11716 | The number 6 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 6 ∈ ℂ | ||
Theorem | 7nn 11717 | 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 7 ∈ ℕ | ||
Theorem | 7re 11718 | The number 7 is real. (Contributed by NM, 27-May-1999.) |
⊢ 7 ∈ ℝ | ||
Theorem | 7cn 11719 | The number 7 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 7 ∈ ℂ | ||
Theorem | 8nn 11720 | 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 8 ∈ ℕ | ||
Theorem | 8re 11721 | The number 8 is real. (Contributed by NM, 27-May-1999.) |
⊢ 8 ∈ ℝ | ||
Theorem | 8cn 11722 | The number 8 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 8 ∈ ℂ | ||
Theorem | 9nn 11723 | 9 is a positive integer. (Contributed by NM, 21-Oct-2012.) |
⊢ 9 ∈ ℕ | ||
Theorem | 9re 11724 | The number 9 is real. (Contributed by NM, 27-May-1999.) |
⊢ 9 ∈ ℝ | ||
Theorem | 9cn 11725 | The number 9 is a complex number. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
⊢ 9 ∈ ℂ | ||
Theorem | 0le0 11726 | Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ 0 ≤ 0 | ||
Theorem | 0le2 11727 | The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.) |
⊢ 0 ≤ 2 | ||
Theorem | 2pos 11728 | The number 2 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 2 | ||
Theorem | 2ne0 11729 | The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.) |
⊢ 2 ≠ 0 | ||
Theorem | 3pos 11730 | The number 3 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 3 | ||
Theorem | 3ne0 11731 | The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.) |
⊢ 3 ≠ 0 | ||
Theorem | 4pos 11732 | The number 4 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 4 | ||
Theorem | 4ne0 11733 | The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ 4 ≠ 0 | ||
Theorem | 5pos 11734 | The number 5 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 5 | ||
Theorem | 6pos 11735 | The number 6 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 6 | ||
Theorem | 7pos 11736 | The number 7 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 7 | ||
Theorem | 8pos 11737 | The number 8 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 8 | ||
Theorem | 9pos 11738 | The number 9 is positive. (Contributed by NM, 27-May-1999.) |
⊢ 0 < 9 | ||
This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering). | ||
Theorem | neg1cn 11739 | -1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ -1 ∈ ℂ | ||
Theorem | neg1rr 11740 | -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.) |
⊢ -1 ∈ ℝ | ||
Theorem | neg1ne0 11741 | -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 ≠ 0 | ||
Theorem | neg1lt0 11742 | -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ -1 < 0 | ||
Theorem | negneg1e1 11743 | --1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ --1 = 1 | ||
Theorem | 1pneg1e0 11744 | 1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + -1) = 0 | ||
Theorem | 0m0e0 11745 | 0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (0 − 0) = 0 | ||
Theorem | 1m0e1 11746 | 1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 − 0) = 1 | ||
Theorem | 0p1e1 11747 | 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (0 + 1) = 1 | ||
Theorem | fv0p1e1 11748 | 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 11749 | 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (1 + 0) = 1 | ||
Theorem | 1p1e2 11750 | 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.) |
⊢ (1 + 1) = 2 | ||
Theorem | 2m1e1 11751 | 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 11780. (Contributed by David A. Wheeler, 4-Jan-2017.) |
⊢ (2 − 1) = 1 | ||
Theorem | 1e2m1 11752 | 1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ 1 = (2 − 1) | ||
Theorem | 3m1e2 11753 | 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 11754 | 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
⊢ (4 − 1) = 3 | ||
Theorem | 5m1e4 11755 | 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.) |
⊢ (5 − 1) = 4 | ||
Theorem | 6m1e5 11756 | 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.) |
⊢ (6 − 1) = 5 | ||
Theorem | 7m1e6 11757 | 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.) |
⊢ (7 − 1) = 6 | ||
Theorem | 8m1e7 11758 | 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.) |
⊢ (8 − 1) = 7 | ||
Theorem | 9m1e8 11759 | 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.) |
⊢ (9 − 1) = 8 | ||
Theorem | 2p2e4 11760 | 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 8155 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 11761 | 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 11762 | A number times 2. (Contributed by NM, 16-Oct-2007.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | ||
Theorem | 2timesi 11763 | Two times a number. (Contributed by NM, 1-Aug-1999.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (2 · 𝐴) = (𝐴 + 𝐴) | ||
Theorem | times2i 11764 | A number times 2. (Contributed by NM, 11-May-2004.) |
⊢ 𝐴 ∈ ℂ ⇒ ⊢ (𝐴 · 2) = (𝐴 + 𝐴) | ||
Theorem | 2txmxeqx 11765 | 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 11766 | 2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 / 2) = 1 | ||
Theorem | 2p1e3 11767 | 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (2 + 1) = 3 | ||
Theorem | 1p2e3 11768 | 1 + 2 = 3. For a shorter proof using addcomli 10820, see 1p2e3ALT 11769. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.) |
⊢ (1 + 2) = 3 | ||
Theorem | 1p2e3ALT 11769 | Alternate proof of 1p2e3 11768, shorter but using more axioms. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (1 + 2) = 3 | ||
Theorem | 3p1e4 11770 | 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (3 + 1) = 4 | ||
Theorem | 4p1e5 11771 | 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (4 + 1) = 5 | ||
Theorem | 5p1e6 11772 | 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (5 + 1) = 6 | ||
Theorem | 6p1e7 11773 | 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (6 + 1) = 7 | ||
Theorem | 7p1e8 11774 | 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (7 + 1) = 8 | ||
Theorem | 8p1e9 11775 | 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.) |
⊢ (8 + 1) = 9 | ||
Theorem | 3p2e5 11776 | 3 + 2 = 5. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 2) = 5 | ||
Theorem | 3p3e6 11777 | 3 + 3 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (3 + 3) = 6 | ||
Theorem | 4p2e6 11778 | 4 + 2 = 6. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 2) = 6 | ||
Theorem | 4p3e7 11779 | 4 + 3 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 3) = 7 | ||
Theorem | 4p4e8 11780 | 4 + 4 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (4 + 4) = 8 | ||
Theorem | 5p2e7 11781 | 5 + 2 = 7. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 2) = 7 | ||
Theorem | 5p3e8 11782 | 5 + 3 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 3) = 8 | ||
Theorem | 5p4e9 11783 | 5 + 4 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (5 + 4) = 9 | ||
Theorem | 6p2e8 11784 | 6 + 2 = 8. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 2) = 8 | ||
Theorem | 6p3e9 11785 | 6 + 3 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (6 + 3) = 9 | ||
Theorem | 7p2e9 11786 | 7 + 2 = 9. (Contributed by NM, 11-May-2004.) |
⊢ (7 + 2) = 9 | ||
Theorem | 1t1e1 11787 | 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.) |
⊢ (1 · 1) = 1 | ||
Theorem | 2t1e2 11788 | 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.) |
⊢ (2 · 1) = 2 | ||
Theorem | 2t2e4 11789 | 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.) |
⊢ (2 · 2) = 4 | ||
Theorem | 3t1e3 11790 | 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (3 · 1) = 3 | ||
Theorem | 3t2e6 11791 | 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.) |
⊢ (3 · 2) = 6 | ||
Theorem | 3t3e9 11792 | 3 times 3 equals 9. (Contributed by NM, 11-May-2004.) |
⊢ (3 · 3) = 9 | ||
Theorem | 4t2e8 11793 | 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.) |
⊢ (4 · 2) = 8 | ||
Theorem | 2t0e0 11794 | 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
⊢ (2 · 0) = 0 | ||
Theorem | 4d2e2 11795 | One half of four is two. (Contributed by NM, 3-Sep-1999.) |
⊢ (4 / 2) = 2 | ||
Theorem | 1lt2 11796 | 1 is less than 2. (Contributed by NM, 24-Feb-2005.) |
⊢ 1 < 2 | ||
Theorem | 2lt3 11797 | 2 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 2 < 3 | ||
Theorem | 1lt3 11798 | 1 is less than 3. (Contributed by NM, 26-Sep-2010.) |
⊢ 1 < 3 | ||
Theorem | 3lt4 11799 | 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 3 < 4 | ||
Theorem | 2lt4 11800 | 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
⊢ 2 < 4 |
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