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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement

6.2.13  Decimal arithmetic (cont.)

Theoremdec2dvds 13401 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds 13402 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5dvds2 13403 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec5nprm 13404 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremdec2nprm 13405 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theoremmodxai 13406 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)

Theoremmod2xi 13407 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodxp1i 13408 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmod2xnegi 13409 Version of mod2xi 13407 with a negaive mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)

Theoremmodsubi 13410 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremgcdi 13411 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremgcdmodi 13412 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremdecexp2 13413 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)

Theoremnumexp0 13414 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp1 13415 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexpp1 13416 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremnumexp2x 13417 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)

Theoremdecsplit0b 13418 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit0 13419 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)

Theoremdecsplit1 13420 Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.)
;

Theoremdecsplit 13421 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
; ;

Theoremkaratsuba 13422 The Karatsuba multiplication algorithm. If and are decomposed into two groups of digits of length (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log2 3) instead of n^2 for the "naive" algorithm decmul1c 10431. (Contributed by Mario Carneiro, 16-Jul-2015.)

Theorem2exp4 13423 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp6 13424 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2exp8 13425 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
;;

Theorem2exp16 13426 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
; ;;;;

Theorem3exp3 13427 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem2expltfac 13428 The factorial grows faster than two to the power . (Contributed by Mario Carneiro, 15-Sep-2016.)

6.2.14  Specific prime numbers

Theorem4nprm 13429 4 is not a prime number. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem0 13430* Lemma for prmlem1 13432 and prmlem2 13444. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem1a 13431* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theoremprmlem1 13432 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
;

Theorem5prm 13433 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem6nprm 13434 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem7prm 13435 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)

Theorem8nprm 13436 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem9nprm 13437 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem10nprm 13438 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)

Theorem11prm 13439 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem13prm 13440 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem17prm 13441 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem19prm 13442 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theorem23prm 13443 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;

Theoremprmlem2 13444 Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows us to cover the numbers less than . Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to , from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 13457).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

;;                                          ;        ;        ;        ;        ;

Theorem37prm 13445 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem43prm 13446 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem83prm 13447 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorem139prm 13448 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem163prm 13449 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem317prm 13450 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

Theorem631prm 13451 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;

6.2.15  Very large primes

Theorem1259lem1 13452 Lemma for 1259prm 13457. Calculate a power mod. In decimal, we calculate and in this lemma. (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;

Theorem1259lem2 13453 Lemma for 1259prm 13457. Calculate a power mod. In decimal, we calculate . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;

Theorem1259lem3 13454 Lemma for 1259prm 13457. Calculate a power mod. In decimal, we calculate and . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem1259lem4 13455 Lemma for 1259prm 13457. Calculate a power mod. In decimal, we calculate , , and finally . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem1259lem5 13456 Lemma for 1259prm 13457. Calculate the GCD of with . (Contributed by Mario Carneiro, 22-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem1259prm 13457 1259 is a prime number. (Contributed by Mario Carneiro, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem2503lem1 13458 Lemma for 2503prm 13461. Calculate a power mod. In decimal, we calculate . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ; ;;;

Theorem2503lem2 13459 Lemma for 2503prm 13461. Calculate a power mod. We calculate , , , , , , , , and finally . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem2503lem3 13460 Lemma for 2503prm 13461. Calculate the GCD of with . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;

Theorem2503prm 13461 2503 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem4001lem1 13462 Lemma for 4001prm 13466. Calculate a power mod. In decimal, we calculate , , , , and . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;; ;;

Theorem4001lem2 13463 Lemma for 2503prm 13461. Calculate a power mod. In decimal, we calculate and . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;; ;;;

Theorem4001lem3 13464 Lemma for 4001prm 13466. Calculate a power mod. In decimal, we calculate and finally . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;

Theorem4001lem4 13465 Lemma for 4001prm 13466. Calculate the GCD of with . (Contributed by Mario Carneiro, 3-Mar-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
;;;       ;;

Theorem4001prm 13466 4001 is a prime number. (Contributed by Mario Carneiro, 3-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;;;

PART 7  BASIC STRUCTURES

7.1  Extensible structures

7.1.1  Basic definitions

An "extensible structure" is a function (set of ordered pairs) on a finite (and not necessarily sequential) subset of , used to define a specific group, ring, poset, etc. The function's argument is the index of a structure component (such as for the base set of a group), and its value is the component (such as the base set). A group will have at least two components (base set and operation), although it can be further specialized by adding other components such as a multiplicative operation for rings (and still remain a group per our definition). Thus, every ring is also a group. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them.

Syntaxcstr 13467 Extend class notation with the class of structures with components numbered below .
Struct

Syntaxcnx 13468 Extend class notation with the structure component index extractor.

Syntaxcsts 13469 Set components of a structure.
sSet

Syntaxcslot 13470 Extend class notation with the slot function.
Slot

Syntaxcbs 13471 Extend class notation with the class of all base set extractors.

Syntaxcress 13472 Extend class notation with the extensible structure builder restriction operator.
s

Definitiondf-struct 13473* Define a structure with components in . This is not a requirement for groups, posets, etc., but it is a useful assumption for component extraction theorems. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Definitiondf-ndx 13474 Define the structure component index extractor. See theorem ndxarg 13491 to understand its purpose. The restriction to allows to exist as a set, since is a proper class. In principle, we could have chosen or (if we revise all structure component definitions such as df-base 13476) another set such as the natural ordinal numbers (df-om 4848). (Contributed by NM, 4-Sep-2011.)

Definitiondf-slot 13475* Define slot extractor for posets and related structures. Note that the function argument can be any set, although it is meaningful only if it is a member of (df-poset 14405) when used for posets or of (df-grp 14814) when used from groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
Slot

Definitiondf-base 13476 Define the base set (also called underlying set or carrier set) extractor for posets and related structures. (Contributed by NM, 4-Sep-2011.) (Revised by Mario Carneiro, 14-Aug-2015.)
Slot

Definitiondf-sets 13477* Set one or more components of a structure. This function is useful for taking an existing structure and "overriding" one of its components. For example, df-ress 13478 adjusts the base set to match its second argument, which has the effect of making subgroups, subspaces, subrings etc. from the original structures. Or df-mgp 15651, which takes a ring and overrides its addition operation with the multiplicative operation, so that we can consider the "multiplicative group" using group and monoid theorems, which expect the operation to be in the slot instead of the slot. (Contributed by Mario Carneiro, 1-Dec-2014.)
sSet

Definitiondf-ress 13478* Define a multifunction restriction operator for extensible structures, which can be used to turn statements about rings into statements about subrings, modules into submodules, etc. This definition knows nothing about individual structures and merely truncates the set while leaving operators alone; individual kinds of structures will need to handle this behavior, by ignoring operators' values outside the range (like ), defining a function using the base set and applying that (like ), or explicitly truncating the slot before use (like ).

(Credit for this operator goes to Mario Carneiro).

See ressbas 13521 for the altered base set, and resslem 13524 (subrg0 15877, ressplusg 13573, subrg1 15880, ressmulr 13584) for the (un)altered other operations. (Contributed by Stefan O'Rear, 29-Nov-2014.)

s sSet

Theorembrstruct 13479 The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremisstruct2 13480 The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremisstruct 13481 The property of being a structure with components in . (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstructcnvcnv 13482 Two ways to express the relational part of a structure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstructfun 13483 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremstructfn 13484 Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.)
Struct

Theoremslotfn 13485 A slot is a function on sets, treated as structures. (Contributed by Mario Carneiro, 22-Sep-2015.)
Slot

Theoremstrfvnd 13486 Deduction version of strfvn 13488. (Contributed by Mario Carneiro, 15-Nov-2014.)
Slot

Theoremwunndx 13487 Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni

Theoremstrfvn 13488 Value of a structure component extractor . Normally, is a defined constant symbol such as (df-base 13476) and is a fixed integer such as . is a structure, i.e. a specific member of a class of structures such as (df-poset 14405) where .

Note: Normally, this theorem shouldn't be used outside of this section, because it requires hard-coded index values. Instead, use strfv 13503. (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2013.) (New usage is discouraged.)

Slot

Theoremstrfvss 13489 A structure component extractor produces a value which is contained in a set dependent on , but not . This is sometimes useful for showing sethood. (Contributed by Mario Carneiro, 15-Aug-2015.)
Slot

Theoremwunstr 13490 Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
Slot        WUni

Theoremndxarg 13491 Get the numeric argument from a defined structure component extractor such as df-base 13476. (Contributed by Mario Carneiro, 6-Oct-2013.)
Slot

Theoremndxid 13492 A structure component extractor is defined by its own index. This theorem, together with strfv 13503 below, is useful for avoiding direct reference to the hard-coded numeric index in component extractor definitions, such as the in df-base 13476 and the in df-ple 13551, making it easier to change should the need arise. For example, we can refer to a specific poset with base set and order relation using rather than . The latter, while shorter to state, requires revision if we later change to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.)
Slot               Slot

Theoremreldmsets 13493 The structure override operator is a proper operator. (Contributed by Stefan O'Rear, 29-Jan-2015.)
sSet

Theoremsetsvalg 13494 Value of the structure replacement function. (Contributed by Mario Carneiro, 30-Apr-2015.)
sSet

Theoremsetsval 13495 Value of the structure replacement function. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
sSet

Theoremwunsets 13496 Closure of structure replacement in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.)
WUni                     sSet

Theoremsetsres 13497 The structure replacement function does not affect the value of away from . (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
sSet

Theoremsetsabs 13498 Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
sSet sSet sSet

Theoremsetscom 13499 Component-setting is commutative when the x-values are different. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
sSet sSet sSet sSet

Theoremstrfvd 13500 Deduction version of strfv 13503. (Contributed by Mario Carneiro, 15-Nov-2014.)
Slot

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