Theorem List for Intuitionistic Logic Explorer - 8301-8400 *Has distinct variable
group(s)
Type | Label | Description |
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Theorem | zltlem1 8301 |
Integer ordering relation. (Contributed by NM, 13-Nov-2004.)
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Theorem | zgt0ge1 8302 |
An integer greater than
is greater than or equal to .
(Contributed by AV, 14-Oct-2018.)
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Theorem | nnleltp1 8303 |
Positive integer ordering relation. (Contributed by NM, 13-Aug-2001.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nnltp1le 8304 |
Positive integer ordering relation. (Contributed by NM, 19-Aug-2001.)
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Theorem | nnaddm1cl 8305 |
Closure of addition of positive integers minus one. (Contributed by NM,
6-Aug-2003.) (Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nn0ltp1le 8306 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Dec-2002.) (Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | nn0leltp1 8307 |
Nonnegative integer ordering relation. (Contributed by Raph Levien,
10-Apr-2004.)
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Theorem | nn0ltlem1 8308 |
Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004.)
(Proof shortened by Mario Carneiro, 16-May-2014.)
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Theorem | znn0sub 8309 |
The nonnegative difference of integers is a nonnegative integer.
(Generalization of nn0sub 8310.) (Contributed by NM, 14-Jul-2005.)
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Theorem | nn0sub 8310 |
Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.)
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Theorem | nn0n0n1ge2 8311 |
A nonnegative integer which is neither 0 nor 1 is greater than or equal to
2. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
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Theorem | elz2 8312* |
Membership in the set of integers. Commonly used in constructions of
the integers as equivalence classes under subtraction of the positive
integers. (Contributed by Mario Carneiro, 16-May-2014.)
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Theorem | dfz2 8313 |
Alternative definition of the integers, based on elz2 8312.
(Contributed
by Mario Carneiro, 16-May-2014.)
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Theorem | nn0sub2 8314 |
Subtraction of nonnegative integers. (Contributed by NM, 4-Sep-2005.)
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Theorem | zapne 8315 |
Apartness is equivalent to not equal for integers. (Contributed by Jim
Kingdon, 14-Mar-2020.)
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Theorem | zdceq 8316 |
Equality of integers is decidable. (Contributed by Jim Kingdon,
14-Mar-2020.)
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DECID
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Theorem | zdcle 8317 |
Integer is
decidable. (Contributed by Jim Kingdon, 7-Apr-2020.)
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DECID |
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Theorem | zdclt 8318 |
Integer is
decidable. (Contributed by Jim Kingdon, 1-Jun-2020.)
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DECID |
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Theorem | zltlen 8319 |
Integer 'Less than' expressed in terms of 'less than or equal to'. Also
see ltleap 7621 which is a similar result for real numbers.
(Contributed by
Jim Kingdon, 14-Mar-2020.)
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Theorem | nn0n0n1ge2b 8320 |
A nonnegative integer is neither 0 nor 1 if and only if it is greater than
or equal to 2. (Contributed by Alexander van der Vekens, 17-Jan-2018.)
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Theorem | nn0lt10b 8321 |
A nonnegative integer less than is .
(Contributed by Paul
Chapman, 22-Jun-2011.)
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Theorem | nn0lt2 8322 |
A nonnegative integer less than 2 must be 0 or 1. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
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Theorem | nn0lem1lt 8323 |
Nonnegative integer ordering relation. (Contributed by NM,
21-Jun-2005.)
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Theorem | nnlem1lt 8324 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
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Theorem | nnltlem1 8325 |
Positive integer ordering relation. (Contributed by NM, 21-Jun-2005.)
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Theorem | nnm1ge0 8326 |
A positive integer decreased by 1 is greater than or equal to 0.
(Contributed by AV, 30-Oct-2018.)
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Theorem | nn0ge0div 8327 |
Division of a nonnegative integer by a positive number is not negative.
(Contributed by Alexander van der Vekens, 14-Apr-2018.)
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Theorem | zdiv 8328* |
Two ways to express " divides .
(Contributed by NM,
3-Oct-2008.)
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Theorem | zdivadd 8329 |
Property of divisibility: if divides
and then it divides
. (Contributed by NM, 3-Oct-2008.)
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Theorem | zdivmul 8330 |
Property of divisibility: if divides
then it divides
. (Contributed by NM, 3-Oct-2008.)
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Theorem | zextle 8331* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
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Theorem | zextlt 8332* |
An extensionality-like property for integer ordering. (Contributed by
NM, 29-Oct-2005.)
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Theorem | recnz 8333 |
The reciprocal of a number greater than 1 is not an integer. (Contributed
by NM, 3-May-2005.)
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Theorem | btwnnz 8334 |
A number between an integer and its successor is not an integer.
(Contributed by NM, 3-May-2005.)
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Theorem | gtndiv 8335 |
A larger number does not divide a smaller positive integer. (Contributed
by NM, 3-May-2005.)
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Theorem | halfnz 8336 |
One-half is not an integer. (Contributed by NM, 31-Jul-2004.)
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Theorem | prime 8337* |
Two ways to express " is a prime number (or 1)." (Contributed by
NM, 4-May-2005.)
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Theorem | msqznn 8338 |
The square of a nonzero integer is a positive integer. (Contributed by
NM, 2-Aug-2004.)
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Theorem | zneo 8339 |
No even integer equals an odd integer (i.e. no integer can be both even
and odd). Exercise 10(a) of [Apostol] p.
28. (Contributed by NM,
31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)
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Theorem | nneoor 8340 |
A positive integer is even or odd. (Contributed by Jim Kingdon,
15-Mar-2020.)
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Theorem | nneo 8341 |
A positive integer is even or odd but not both. (Contributed by NM,
1-Jan-2006.) (Proof shortened by Mario Carneiro, 18-May-2014.)
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Theorem | nneoi 8342 |
A positive integer is even or odd but not both. (Contributed by NM,
20-Aug-2001.)
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Theorem | zeo 8343 |
An integer is even or odd. (Contributed by NM, 1-Jan-2006.)
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Theorem | zeo2 8344 |
An integer is even or odd but not both. (Contributed by Mario Carneiro,
12-Sep-2015.)
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Theorem | peano2uz2 8345* |
Second Peano postulate for upper integers. (Contributed by NM,
3-Oct-2004.)
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Theorem | peano5uzti 8346* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 25-Jul-2013.)
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Theorem | peano5uzi 8347* |
Peano's inductive postulate for upper integers. (Contributed by NM,
6-Jul-2005.) (Revised by Mario Carneiro, 3-May-2014.)
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Theorem | dfuzi 8348* |
An expression for the upper integers that start at that is
analogous to dfnn2 7916 for positive integers. (Contributed by NM,
6-Jul-2005.) (Proof shortened by Mario Carneiro, 3-May-2014.)
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Theorem | uzind 8349* |
Induction on the upper integers that start at . The first four
hypotheses give us the substitution instances we need; the last two are
the basis and the induction step. (Contributed by NM, 5-Jul-2005.)
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Theorem | uzind2 8350* |
Induction on the upper integers that start after an integer .
The first four hypotheses give us the substitution instances we need;
the last two are the basis and the induction step. (Contributed by NM,
25-Jul-2005.)
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Theorem | uzind3 8351* |
Induction on the upper integers that start at an integer . The
first four hypotheses give us the substitution instances we need, and
the last two are the basis and the induction step. (Contributed by NM,
26-Jul-2005.)
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Theorem | nn0ind 8352* |
Principle of Mathematical Induction (inference schema) on nonnegative
integers. The first four hypotheses give us the substitution instances
we need; the last two are the basis and the induction step.
(Contributed by NM, 13-May-2004.)
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Theorem | fzind 8353* |
Induction on the integers from to
inclusive . The first
four hypotheses give us the substitution instances we need; the last two
are the basis and the induction step. (Contributed by Paul Chapman,
31-Mar-2011.)
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Theorem | fnn0ind 8354* |
Induction on the integers from to
inclusive . The first
four hypotheses give us the substitution instances we need; the last two
are the basis and the induction step. (Contributed by Paul Chapman,
31-Mar-2011.)
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Theorem | nn0ind-raph 8355* |
Principle of Mathematical Induction (inference schema) on nonnegative
integers. The first four hypotheses give us the substitution instances
we need; the last two are the basis and the induction step. Raph Levien
remarks: "This seems a bit painful. I wonder if an explicit
substitution version would be easier." (Contributed by Raph
Levien,
10-Apr-2004.)
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Theorem | zindd 8356* |
Principle of Mathematical Induction on all integers, deduction version.
The first five hypotheses give the substitutions; the last three are the
basis, the induction, and the extension to negative numbers.
(Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario
Carneiro, 4-Jan-2017.)
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Theorem | btwnz 8357* |
Any real number can be sandwiched between two integers. Exercise 2 of
[Apostol] p. 28. (Contributed by NM,
10-Nov-2004.)
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Theorem | nn0zd 8358 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | nnzd 8359 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zred 8360 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zcnd 8361 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | znegcld 8362 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | peano2zd 8363 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | zaddcld 8364 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zsubcld 8365 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zmulcld 8366 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zadd2cl 8367 |
Increasing an integer by 2 results in an integer. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
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3.4.9 Decimal arithmetic
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Syntax | cdc 8368 |
Constant used for decimal constructor.
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Definition | df-dec 8369 |
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base 10. For example,
;;; ;;; ;;;. (Contributed by
Mario Carneiro, 17-Apr-2015.)
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Theorem | deceq1 8370 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | deceq2 8371 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | deceq1i 8372 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | deceq2i 8373 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | deceq12i 8374 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | numnncl 8375 |
Closure for a numeral (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | num0u 8376 |
Add a zero in the units place. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | num0h 8377 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | numcl 8378 |
Closure for a decimal integer (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | numsuc 8379 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | decnncl 8380 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
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Theorem | deccl 8381 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
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Theorem | dec0u 8382 |
Add a zero in the units place. (Contributed by Mario Carneiro,
17-Apr-2015.)
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Theorem | dec0h 8383 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
17-Apr-2015.)
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Theorem | numnncl2 8384 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 9-Mar-2015.)
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Theorem | decnncl2 8385 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | numlt 8386 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | numltc 8387 |
Comparing two decimal integers (unequal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | declt 8388 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 17-Apr-2015.)
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Theorem | decltc 8389 |
Comparing two decimal integers (unequal higher places). (Contributed
by Mario Carneiro, 18-Feb-2014.)
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Theorem | decsuc 8390 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 17-Apr-2015.)
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Theorem | numlti 8391 |
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | declti 8392 |
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | numsucc 8393 |
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | decsucc 8394 |
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | 1e0p1 8395 |
The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
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Theorem | dec10p 8396 |
Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
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Theorem | dec10 8397 |
The decimal form of 10. NB: In our presentations of large numbers later
on, we will use our symbol for 10 at the highest digits when advantageous,
because we can use this theorem to convert back to "long form"
(where each
digit is in the range 0-9) with no extra effort. However, we
cannot do
this for lower digits while maintaining the ease of use of the decimal
system, since it requires nontrivial number knowledge (more than just
equality theorems) to convert back. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | numma 8398 |
Perform a multiply-add of two decimal integers and against
a fixed multiplicand (no carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | nummac 8399 |
Perform a multiply-add of two decimal integers and against
a fixed multiplicand (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | numma2c 8400 |
Perform a multiply-add of two decimal integers and against
a fixed multiplicand (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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