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Theorem List for Metamath Proof Explorer - 11701-11800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-3 11701 Define the number 3. (Contributed by NM, 27-May-1999.)
3 = (2 + 1)

Definitiondf-4 11702 Define the number 4. (Contributed by NM, 27-May-1999.)
4 = (3 + 1)

Definitiondf-5 11703 Define the number 5. (Contributed by NM, 27-May-1999.)
5 = (4 + 1)

Definitiondf-6 11704 Define the number 6. (Contributed by NM, 27-May-1999.)
6 = (5 + 1)

Definitiondf-7 11705 Define the number 7. (Contributed by NM, 27-May-1999.)
7 = (6 + 1)

Definitiondf-8 11706 Define the number 8. (Contributed by NM, 27-May-1999.)
8 = (7 + 1)

Definitiondf-9 11707 Define the number 9. (Contributed by NM, 27-May-1999.)
9 = (8 + 1)

Theorem0ne1 11708 Zero is different from one (the commuted form is the axiom ax-1ne0 10605). (Contributed by David A. Wheeler, 8-Dec-2018.)
0 ≠ 1

Theorem1m1e0 11709 One minus one equals zero. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 − 1) = 0

Theorem2nn 11710 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem2re 11711 The number 2 is real. (Contributed by NM, 27-May-1999.)
2 ∈ ℝ

Theorem2cn 11712 The number 2 is a complex number. (Contributed by NM, 30-Jul-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.)
2 ∈ ℂ

Theorem2cnALT 11713 Alternate proof of 2cn 11712. Shorter but uses more axioms. Similar proofs are possible for 3cn 11718, ... , 9cn 11737. (Contributed by NM, 30-Jul-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
2 ∈ ℂ

Theorem2ex 11714 The number 2 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
2 ∈ V

Theorem2cnd 11715 The number 2 is a complex number, deduction form. (Contributed by David A. Wheeler, 8-Dec-2018.)
(𝜑 → 2 ∈ ℂ)

Theorem3nn 11716 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem3re 11717 The number 3 is real. (Contributed by NM, 27-May-1999.)
3 ∈ ℝ

Theorem3cn 11718 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

Theorem3ex 11719 The number 3 is a set. (Contributed by David A. Wheeler, 8-Dec-2018.)
3 ∈ V

Theorem4nn 11720 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem4re 11721 The number 4 is real. (Contributed by NM, 27-May-1999.)
4 ∈ ℝ

Theorem4cn 11722 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

Theorem5nn 11723 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem5re 11724 The number 5 is real. (Contributed by NM, 27-May-1999.)
5 ∈ ℝ

Theorem5cn 11725 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

Theorem6nn 11726 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem6re 11727 The number 6 is real. (Contributed by NM, 27-May-1999.)
6 ∈ ℝ

Theorem6cn 11728 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

Theorem7nn 11729 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem7re 11730 The number 7 is real. (Contributed by NM, 27-May-1999.)
7 ∈ ℝ

Theorem7cn 11731 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

Theorem8nn 11732 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem8re 11733 The number 8 is real. (Contributed by NM, 27-May-1999.)
8 ∈ ℝ

Theorem8cn 11734 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

Theorem9nn 11735 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem9re 11736 The number 9 is real. (Contributed by NM, 27-May-1999.)
9 ∈ ℝ

Theorem9cn 11737 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

Theorem0le0 11738 Zero is nonnegative. (Contributed by David A. Wheeler, 7-Jul-2016.)
0 ≤ 0

Theorem0le2 11739 The number 0 is less than or equal to 2. (Contributed by David A. Wheeler, 7-Dec-2018.)
0 ≤ 2

Theorem2pos 11740 The number 2 is positive. (Contributed by NM, 27-May-1999.)
0 < 2

Theorem2ne0 11741 The number 2 is nonzero. (Contributed by NM, 9-Nov-2007.)
2 ≠ 0

Theorem3pos 11742 The number 3 is positive. (Contributed by NM, 27-May-1999.)
0 < 3

Theorem3ne0 11743 The number 3 is nonzero. (Contributed by FL, 17-Oct-2010.) (Proof shortened by Andrew Salmon, 7-May-2011.)
3 ≠ 0

Theorem4pos 11744 The number 4 is positive. (Contributed by NM, 27-May-1999.)
0 < 4

Theorem4ne0 11745 The number 4 is nonzero. (Contributed by David A. Wheeler, 5-Dec-2018.)
4 ≠ 0

Theorem5pos 11746 The number 5 is positive. (Contributed by NM, 27-May-1999.)
0 < 5

Theorem6pos 11747 The number 6 is positive. (Contributed by NM, 27-May-1999.)
0 < 6

Theorem7pos 11748 The number 7 is positive. (Contributed by NM, 27-May-1999.)
0 < 7

Theorem8pos 11749 The number 8 is positive. (Contributed by NM, 27-May-1999.)
0 < 8

Theorem9pos 11750 The number 9 is positive. (Contributed by NM, 27-May-1999.)
0 < 9

5.4.4  Some properties of specific numbers

This section includes specific theorems about one-digit natural numbers (membership, addition, subtraction, multiplication, division, ordering).

Theoremneg1cn 11751 -1 is a complex number. (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 11752 -1 is a real number. (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 11753 -1 is nonzero. (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 11754 -1 is less than 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremnegneg1e1 11755 --1 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 11756 1 + -1 is 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 11757 0 minus 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 11758 1 - 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 11759 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theoremfv0p1e1 11760 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))

Theorem1p0e1 11761 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 11762 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 11763 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 11792. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 11764 1 = 2 - 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 11765 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

Theorem4m1e3 11766 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
(4 − 1) = 3

Theorem5m1e4 11767 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
(5 − 1) = 4

Theorem6m1e5 11768 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5

Theorem7m1e6 11769 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6

Theorem8m1e7 11770 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7

Theorem9m1e8 11771 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8

Theorem2p2e4 11772 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 8163 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

Theorem2times 11773 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 11774 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 11775 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 11776 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

Theorem2txmxeqx 11777 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)

Theorem2div2e1 11778 2 divided by 2 is 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

Theorem2p1e3 11779 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3

Theorem1p2e3 11780 1 + 2 = 3. For a shorter proof using addcomli 10831, see 1p2e3ALT 11781. (Contributed by David A. Wheeler, 8-Dec-2018.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 12-Dec-2022.)
(1 + 2) = 3

Theorem1p2e3ALT 11781 Alternate proof of 1p2e3 11780, shorter but using more axioms. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(1 + 2) = 3

Theorem3p1e4 11782 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4

Theorem4p1e5 11783 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5

Theorem5p1e6 11784 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6

Theorem6p1e7 11785 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7

Theorem7p1e8 11786 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8

Theorem8p1e9 11787 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9

Theorem3p2e5 11788 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5

Theorem3p3e6 11789 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6

Theorem4p2e6 11790 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6

Theorem4p3e7 11791 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7

Theorem4p4e8 11792 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8

Theorem5p2e7 11793 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7

Theorem5p3e8 11794 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8

Theorem5p4e9 11795 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem6p2e8 11796 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 11797 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem7p2e9 11798 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem1t1e1 11799 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 11800 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

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