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Theorem | demoivreALT 11801 | Alternate proof of demoivre 11800. It is longer but does not use the exponential function. This is Metamath 100 proof #17. (Contributed by Steve Rodriguez, 10-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Syntax | ctau 11802 |
Extend class notation to include the constant tau, ![]() |
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Definition | df-tau 11803 |
Define the circle constant tau, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eirraplem 11804* | Lemma for eirrap 11805. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.) |
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Theorem | eirrap 11805 |
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Theorem | eirr 11806 |
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Theorem | egt2lt3 11807 |
Euler's constant ![]() |
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Theorem | epos 11808 |
Euler's constant ![]() |
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Theorem | epr 11809 |
Euler's constant ![]() |
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Theorem | ene0 11810 |
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Theorem | eap0 11811 |
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Theorem | ene1 11812 |
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Theorem | eap1 11813 |
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This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 11814 | Extend the definition of a class to include the divides relation. See df-dvds 11815. |
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Definition | df-dvds 11815* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | divides 11816* |
Define the divides relation. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsval2 11817 | One nonzero integer divides another integer if and only if their quotient is an integer. (Contributed by Jeff Hankins, 29-Sep-2013.) |
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Theorem | dvdsval3 11818 | One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in [ApostolNT] p. 106. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 15-Jul-2014.) |
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Theorem | dvdszrcl 11819 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | dvdsmod0 11820 | If a positive integer divides another integer, then the remainder upon division is zero. (Contributed by AV, 3-Mar-2022.) |
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Theorem | p1modz1 11821 | If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.) |
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Theorem | dvdsmodexp 11822 | If a positive integer divides another integer, this other integer is equal to its positive powers modulo the positive integer. (Formerly part of the proof for fermltl 12254). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.) |
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Theorem | nndivdvds 11823 | Strong form of dvdsval2 11817 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | nndivides 11824* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
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Theorem | dvdsdc 11825 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
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Theorem | moddvds 11826 |
Two ways to say ![]() ![]() ![]() ![]() |
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Theorem | modm1div 11827 | An integer greater than one divides another integer minus one iff the second integer modulo the first integer is one. (Contributed by AV, 30-May-2023.) |
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Theorem | dvds0lem 11828 |
A lemma to assist theorems of ![]() |
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Theorem | dvds1lem 11829* |
A lemma to assist theorems of ![]() |
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Theorem | dvds2lem 11830* |
A lemma to assist theorems of ![]() |
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Theorem | iddvds 11831 | An integer divides itself. Theorem 1.1(a) in [ApostolNT] p. 14 (reflexive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 1dvds 11832 | 1 divides any integer. Theorem 1.1(f) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds0 11833 | Any integer divides 0. Theorem 1.1(g) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | negdvdsb 11834 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsnegb 11835 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | absdvdsb 11836 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsabsb 11837 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 0dvds 11838 | Only 0 is divisible by 0. Theorem 1.1(h) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | zdvdsdc 11839 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | dvdsmul1 11840 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmul2 11841 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | iddvdsexp 11842 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
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Theorem | muldvds1 11843 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | muldvds2 11844 | If a product divides an integer, so does one of its factors. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmul 11845 | Multiplication by a constant maintains the divides relation. Theorem 1.1(d) in [ApostolNT] p. 14 (multiplication property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulc 11846 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmulr 11847 | Cancellation law for the divides relation. Theorem 1.1(e) in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmulcr 11848 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | summodnegmod 11849 | The sum of two integers modulo a positive integer equals zero iff the first of the two integers equals the negative of the other integer modulo the positive integer. (Contributed by AV, 25-Jul-2021.) |
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Theorem | modmulconst 11850 | Constant multiplication in a modulo operation, see theorem 5.3 in [ApostolNT] p. 108. (Contributed by AV, 21-Jul-2021.) |
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Theorem | dvds2ln 11851 | If an integer divides each of two other integers, it divides any linear combination of them. Theorem 1.1(c) in [ApostolNT] p. 14 (linearity property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2add 11852 | If an integer divides each of two other integers, it divides their sum. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2sub 11853 | If an integer divides each of two other integers, it divides their difference. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2subd 11854 | Deduction form of dvds2sub 11853. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdstr 11855 | The divides relation is transitive. Theorem 1.1(b) in [ApostolNT] p. 14 (transitive property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvds2addd 11856 | Deduction form of dvds2add 11852. (Contributed by SN, 21-Aug-2024.) |
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Theorem | dvdstrd 11857 | The divides relation is transitive, a deduction version of dvdstr 11855. (Contributed by metakunt, 12-May-2024.) |
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Theorem | dvdsmultr1 11858 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | dvdsmultr1d 11859 | Natural deduction form of dvdsmultr1 11858. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdsmultr2 11860 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | ordvdsmul 11861 | If an integer divides either of two others, it divides their product. (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 17-Jul-2014.) |
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Theorem | dvdssub2 11862 | If an integer divides a difference, then it divides one term iff it divides the other. (Contributed by Mario Carneiro, 13-Jul-2014.) |
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Theorem | dvdsadd 11863 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 13-Jul-2014.) |
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Theorem | dvdsaddr 11864 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssub 11865 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssubr 11866 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdsadd2b 11867 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
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Theorem | dvdsaddre2b 11868 |
Adding a multiple of the base does not affect divisibility. Variant of
dvdsadd2b 11867 only requiring ![]() |
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Theorem | dvdslelemd 11869 | Lemma for dvdsle 11870. (Contributed by Jim Kingdon, 8-Nov-2021.) |
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Theorem | dvdsle 11870 |
The divisors of a positive integer are bounded by it. The proof does
not use ![]() |
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Theorem | dvdsleabs 11871 | The divisors of a nonzero integer are bounded by its absolute value. Theorem 1.1(i) in [ApostolNT] p. 14 (comparison property of the divides relation). (Contributed by Paul Chapman, 21-Mar-2011.) (Proof shortened by Fan Zheng, 3-Jul-2016.) |
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Theorem | dvdsleabs2 11872 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
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Theorem | dvdsabseq 11873 | If two integers divide each other, they must be equal, up to a difference in sign. Theorem 1.1(j) in [ApostolNT] p. 14. (Contributed by Mario Carneiro, 30-May-2014.) (Revised by AV, 7-Aug-2021.) |
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Theorem | dvdseq 11874 | If two nonnegative integers divide each other, they must be equal. (Contributed by Mario Carneiro, 30-May-2014.) (Proof shortened by AV, 7-Aug-2021.) |
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Theorem | divconjdvds 11875 |
If a nonzero integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsdivcl 11876* |
The complement of a divisor of ![]() ![]() |
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Theorem | dvdsflip 11877* | An involution of the divisors of a number. (Contributed by Stefan O'Rear, 12-Sep-2015.) (Proof shortened by Mario Carneiro, 13-May-2016.) |
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Theorem | dvdsssfz1 11878* | The set of divisors of a number is a subset of a finite set. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | dvds1 11879 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
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Theorem | alzdvds 11880* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsext 11881* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | fzm1ndvds 11882 |
No number between ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzo0dvdseq 11883 |
Zero is the only one of the first ![]() ![]() |
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Theorem | fzocongeq 11884 | Two different elements of a half-open range are not congruent mod its length. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | addmodlteqALT 11885 | Two nonnegative integers less than the modulus are equal iff the sums of these integer with another integer are equal modulo the modulus. Shorter proof of addmodlteq 10418 based on the "divides" relation. (Contributed by AV, 14-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | dvdsfac 11886 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
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Theorem | dvdsexp 11887 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
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Theorem | dvdsmod 11888 |
Any number ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | mulmoddvds 11889 | If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018.) |
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Theorem | 3dvdsdec 11890 |
A decimal number is divisible by three iff the sum of its two
"digits"
is divisible by three. The term "digits" in its narrow sense
is only
correct if ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 3dvds2dec 11891 |
A decimal number is divisible by three iff the sum of its three
"digits"
is divisible by three. The term "digits" in its narrow sense
is only
correct if ![]() ![]() ![]() ![]() ![]() ![]() |
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The set | ||
Theorem | evenelz 11892 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11819. (Contributed by AV, 22-Jun-2021.) |
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Theorem | zeo3 11893 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeoxor 11894 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
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Theorem | zeo4 11895 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeneo 11896 | No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 9374 follows immediately from the fact that a contradiction implies anything, see pm2.21i 647. (Contributed by AV, 22-Jun-2021.) |
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Theorem | odd2np1lem 11897* | Lemma for odd2np1 11898. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | odd2np1 11898* | An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | even2n 11899* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
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Theorem | oddm1even 11900 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
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