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Type | Label | Description |
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Statement | ||
Definition | df-tau 11801 |
Define the circle constant tau, ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | eirraplem 11802* | Lemma for eirrap 11803. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.) |
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Theorem | eirrap 11803 |
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Theorem | eirr 11804 |
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Theorem | egt2lt3 11805 |
Euler's constant ![]() |
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Theorem | epos 11806 |
Euler's constant ![]() |
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Theorem | epr 11807 |
Euler's constant ![]() |
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Theorem | ene0 11808 |
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Theorem | eap0 11809 |
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Theorem | ene1 11810 |
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Theorem | eap1 11811 |
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This part introduces elementary number theory, in particular the elementary properties of divisibility and elementary prime number theory. | ||
Syntax | cdvds 11812 | Extend the definition of a class to include the divides relation. See df-dvds 11813. |
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Definition | df-dvds 11813* | Define the divides relation, see definition in [ApostolNT] p. 14. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | divides 11814* |
Define the divides relation. ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsval2 11815 | 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 11816 | 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 11817 | Reverse closure for the divisibility relation. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
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Theorem | dvdsmod0 11818 | 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 11819 | 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 11820 | 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 12252). (Contributed by Mario Carneiro, 28-Feb-2014.) (Revised by AV, 19-Mar-2022.) |
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Theorem | nndivdvds 11821 | Strong form of dvdsval2 11815 for positive integers. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
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Theorem | nndivides 11822* | Definition of the divides relation for positive integers. (Contributed by AV, 26-Jul-2021.) |
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Theorem | dvdsdc 11823 | Divisibility is decidable. (Contributed by Jim Kingdon, 14-Nov-2021.) |
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Theorem | moddvds 11824 |
Two ways to say ![]() ![]() ![]() ![]() |
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Theorem | modm1div 11825 | 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 11826 |
A lemma to assist theorems of ![]() |
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Theorem | dvds1lem 11827* |
A lemma to assist theorems of ![]() |
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Theorem | dvds2lem 11828* |
A lemma to assist theorems of ![]() |
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Theorem | iddvds 11829 | 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 11830 | 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 11831 | 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 11832 | An integer divides another iff its negation does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsnegb 11833 | An integer divides another iff it divides its negation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | absdvdsb 11834 | An integer divides another iff its absolute value does. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsabsb 11835 | An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | 0dvds 11836 | 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 11837 | Divisibility of integers is decidable. (Contributed by Jim Kingdon, 17-Jan-2022.) |
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Theorem | dvdsmul1 11838 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsmul2 11839 | An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | iddvdsexp 11840 | An integer divides a positive integer power of itself. (Contributed by Paul Chapman, 26-Oct-2012.) |
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Theorem | muldvds1 11841 | 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 11842 | 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 11843 | 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 11844 | Multiplication by a constant maintains the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdscmulr 11845 | 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 11846 | Cancellation law for the divides relation. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | summodnegmod 11847 | 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 11848 | 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 11849 | 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 11850 | 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 11851 | 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 11852 | Deduction form of dvds2sub 11851. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdstr 11853 | 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 11854 | Deduction form of dvds2add 11850. (Contributed by SN, 21-Aug-2024.) |
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Theorem | dvdstrd 11855 | The divides relation is transitive, a deduction version of dvdstr 11853. (Contributed by metakunt, 12-May-2024.) |
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Theorem | dvdsmultr1 11856 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | dvdsmultr1d 11857 | Natural deduction form of dvdsmultr1 11856. (Contributed by Stanislas Polu, 9-Mar-2020.) |
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Theorem | dvdsmultr2 11858 | If an integer divides another, it divides a multiple of it. (Contributed by Paul Chapman, 17-Nov-2012.) |
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Theorem | ordvdsmul 11859 | 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 11860 | 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 11861 | 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 11862 | An integer divides another iff it divides their sum. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssub 11863 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdssubr 11864 | An integer divides another iff it divides their difference. (Contributed by Paul Chapman, 31-Mar-2011.) |
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Theorem | dvdsadd2b 11865 | Adding a multiple of the base does not affect divisibility. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
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Theorem | dvdsaddre2b 11866 |
Adding a multiple of the base does not affect divisibility. Variant of
dvdsadd2b 11865 only requiring ![]() |
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Theorem | dvdslelemd 11867 | Lemma for dvdsle 11868. (Contributed by Jim Kingdon, 8-Nov-2021.) |
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Theorem | dvdsle 11868 |
The divisors of a positive integer are bounded by it. The proof does
not use ![]() |
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Theorem | dvdsleabs 11869 | 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 11870 | Transfer divisibility to an order constraint on absolute values. (Contributed by Stefan O'Rear, 24-Sep-2014.) |
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Theorem | dvdsabseq 11871 | 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 11872 | 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 11873 |
If a nonzero integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | dvdsdivcl 11874* |
The complement of a divisor of ![]() ![]() |
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Theorem | dvdsflip 11875* | 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 11876* | 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 11877 | The only nonnegative integer that divides 1 is 1. (Contributed by Mario Carneiro, 2-Jul-2015.) |
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Theorem | alzdvds 11878* | Only 0 is divisible by all integers. (Contributed by Paul Chapman, 21-Mar-2011.) |
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Theorem | dvdsext 11879* | Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
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Theorem | fzm1ndvds 11880 |
No number between ![]() ![]() ![]() ![]() ![]() |
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Theorem | fzo0dvdseq 11881 |
Zero is the only one of the first ![]() ![]() |
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Theorem | fzocongeq 11882 | 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 11883 | 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 10416 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 11884 | A positive integer divides any greater factorial. (Contributed by Paul Chapman, 28-Nov-2012.) |
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Theorem | dvdsexp 11885 | A power divides a power with a greater exponent. (Contributed by Mario Carneiro, 23-Feb-2014.) |
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Theorem | dvdsmod 11886 |
Any number ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | mulmoddvds 11887 | 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 11888 |
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 11889 |
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 11890 | An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11817. (Contributed by AV, 22-Jun-2021.) |
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Theorem | zeo3 11891 | An integer is even or odd. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeoxor 11892 | An integer is even or odd but not both. (Contributed by Jim Kingdon, 10-Nov-2021.) |
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Theorem | zeo4 11893 | An integer is even or odd but not both. (Contributed by AV, 17-Jun-2021.) |
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Theorem | zeneo 11894 | 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 9372 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 11895* | Lemma for odd2np1 11896. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
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Theorem | odd2np1 11896* | 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 11897* | An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
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Theorem | oddm1even 11898 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
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Theorem | oddp1even 11899 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
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Theorem | oexpneg 11900 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) |
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