Theorem List for Intuitionistic Logic Explorer - 9301-9400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | btwnz 9301* |
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 9302 |
A positive integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | nnzd 9303 |
A nonnegative integer is an integer. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zred 9304 |
An integer is a real number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zcnd 9305 |
An integer is a complex number. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | znegcld 9306 |
Closure law for negative integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | peano2zd 9307 |
Deduction from second Peano postulate generalized to integers.
(Contributed by Mario Carneiro, 28-May-2016.)
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Theorem | zaddcld 9308 |
Closure of addition of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zsubcld 9309 |
Closure of subtraction of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zmulcld 9310 |
Closure of multiplication of integers. (Contributed by Mario Carneiro,
28-May-2016.)
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Theorem | zadd2cl 9311 |
Increasing an integer by 2 results in an integer. (Contributed by
Alexander van der Vekens, 16-Sep-2018.)
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Theorem | btwnapz 9312 |
A number between an integer and its successor is apart from any integer.
(Contributed by Jim Kingdon, 6-Jan-2023.)
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# |
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4.4.10 Decimal arithmetic
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Syntax | cdc 9313 |
Constant used for decimal constructor.
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; |
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Definition | df-dec 9314 |
Define the "decimal constructor", which is used to build up
"decimal
integers" or "numeric terms" in base . For example,
;;; ;;; ;;; 1kp2ke3k 13448.
(Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV,
1-Aug-2021.)
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; |
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Theorem | 9p1e10 9315 |
9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by
Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.)
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; |
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Theorem | dfdec10 9316 |
Version of the definition of the "decimal constructor" using ;
instead of the symbol 10. Of course, this statement cannot be used as
definition, because it uses the "decimal constructor".
(Contributed by
AV, 1-Aug-2021.)
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; ;
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Theorem | deceq1 9317 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | deceq2 9318 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | deceq1i 9319 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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; ; |
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Theorem | deceq2i 9320 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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; ; |
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Theorem | deceq12i 9321 |
Equality theorem for the decimal constructor. (Contributed by Mario
Carneiro, 17-Apr-2015.)
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; ; |
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Theorem | numnncl 9322 |
Closure for a numeral (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | num0u 9323 |
Add a zero in the units place. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | num0h 9324 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | numcl 9325 |
Closure for a decimal integer (with units place). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | numsuc 9326 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | deccl 9327 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 10nn 9328 |
10 is a positive integer. (Contributed by NM, 8-Nov-2012.) (Revised by
AV, 6-Sep-2021.)
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; |
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Theorem | 10pos 9329 |
The number 10 is positive. (Contributed by NM, 5-Feb-2007.) (Revised by
AV, 8-Sep-2021.)
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Theorem | 10nn0 9330 |
10 is a nonnegative integer. (Contributed by Mario Carneiro,
19-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 10re 9331 |
The number 10 is real. (Contributed by NM, 5-Feb-2007.) (Revised by AV,
8-Sep-2021.)
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; |
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Theorem | decnncl 9332 |
Closure for a numeral. (Contributed by Mario Carneiro, 17-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | dec0u 9333 |
Add a zero in the units place. (Contributed by Mario Carneiro,
17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | dec0h 9334 |
Add a zero in the higher places. (Contributed by Mario Carneiro,
17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | numnncl2 9335 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 9-Mar-2015.)
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Theorem | decnncl2 9336 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | numlt 9337 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | numltc 9338 |
Comparing two decimal integers (unequal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | le9lt10 9339 |
A "decimal digit" (i.e. a nonnegative integer less than or equal to
9)
is less then 10. (Contributed by AV, 8-Sep-2021.)
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; |
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Theorem | declt 9340 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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; ; |
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Theorem | decltc 9341 |
Comparing two decimal integers (unequal higher places). (Contributed
by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ; ; |
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Theorem | declth 9342 |
Comparing two decimal integers (unequal higher places). (Contributed
by AV, 8-Sep-2021.)
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; ; |
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Theorem | decsuc 9343 |
The successor of a decimal integer (no carry). (Contributed by Mario
Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
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; ; |
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Theorem | 3declth 9344 |
Comparing two decimal integers with three "digits" (unequal higher
places). (Contributed by AV, 8-Sep-2021.)
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;;
;; |
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Theorem | 3decltc 9345 |
Comparing two decimal integers with three "digits" (unequal higher
places). (Contributed by AV, 15-Jun-2021.) (Revised by AV,
6-Sep-2021.)
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;
; ;; ;; |
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Theorem | decle 9346 |
Comparing two decimal integers (equal higher places). (Contributed by
AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
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; ; |
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Theorem | decleh 9347 |
Comparing two decimal integers (unequal higher places). (Contributed by
AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
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; ; |
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Theorem | declei 9348 |
Comparing a digit to a decimal integer. (Contributed by AV,
17-Aug-2021.)
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; |
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Theorem | numlti 9349 |
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | declti 9350 |
Comparing a digit to a decimal integer. (Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | decltdi 9351 |
Comparing a digit to a decimal integer. (Contributed by AV,
8-Sep-2021.)
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; |
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Theorem | numsucc 9352 |
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | decsucc 9353 |
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ; |
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Theorem | 1e0p1 9354 |
The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
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Theorem | dec10p 9355 |
Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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Theorem | numma 9356 |
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 9357 |
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 9358 |
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 | numadd 9359 |
Add two decimal integers and (no
carry). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | numaddc 9360 |
Add two decimal integers and (with
carry). (Contributed
by Mario Carneiro, 18-Feb-2014.)
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Theorem | nummul1c 9361 |
The product of a decimal integer with a number. (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | nummul2c 9362 |
The product of a decimal integer with a number (with carry).
(Contributed by Mario Carneiro, 18-Feb-2014.)
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Theorem | decma 9363 |
Perform a multiply-add of two numerals and against a fixed
multiplicand
(no carry). (Contributed by Mario Carneiro,
18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
; |
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Theorem | decmac 9364 |
Perform a multiply-add of two numerals and against a fixed
multiplicand
(with carry). (Contributed by Mario Carneiro,
18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
;
; |
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Theorem | decma2c 9365 |
Perform a multiply-add of two numerals and against a fixed
multiplier
(with carry). (Contributed by Mario Carneiro,
18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
;
; |
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Theorem | decadd 9366 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
; |
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Theorem | decaddc 9367 |
Add two numerals and
(with carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
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; |
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Theorem | decaddc2 9368 |
Add two numerals and
(with carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
;
; |
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Theorem | decrmanc 9369 |
Perform a multiply-add of two numerals and against a fixed
multiplicand
(no carry). (Contributed by AV, 16-Sep-2021.)
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;
; |
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Theorem | decrmac 9370 |
Perform a multiply-add of two numerals and against a fixed
multiplicand
(with carry). (Contributed by AV, 16-Sep-2021.)
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;
;
; |
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Theorem | decaddm10 9371 |
The sum of two multiples of 10 is a multiple of 10. (Contributed by AV,
30-Jul-2021.)
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; ; ;
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Theorem | decaddi 9372 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
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;
; |
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Theorem | decaddci 9373 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
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;
;
; |
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Theorem | decaddci2 9374 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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;
;
; |
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Theorem | decsubi 9375 |
Difference between a numeral and a nonnegative integer (no
underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV,
6-Sep-2021.)
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;
; |
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Theorem | decmul1 9376 |
The product of a numeral with a number (no carry). (Contributed by
AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | decmul1c 9377 |
The product of a numeral with a number (with carry). (Contributed by
Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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;
;
; |
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Theorem | decmul2c 9378 |
The product of a numeral with a number (with carry). (Contributed by
Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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;
;
; |
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Theorem | decmulnc 9379 |
The product of a numeral with a number (no carry). (Contributed by AV,
15-Jun-2021.)
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; ; |
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Theorem | 11multnc 9380 |
The product of 11 (as numeral) with a number (no carry). (Contributed
by AV, 15-Jun-2021.)
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; ; |
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Theorem | decmul10add 9381 |
A multiplication of a number and a numeral expressed as addition with
first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.)
(Revised by AV, 6-Sep-2021.)
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; ; |
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Theorem | 6p5lem 9382 |
Lemma for 6p5e11 9385 and related theorems. (Contributed by Mario
Carneiro, 19-Apr-2015.)
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; |
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Theorem | 5p5e10 9383 |
5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 6p4e10 9384 |
6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 6p5e11 9385 |
6 + 5 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
AV, 6-Sep-2021.)
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; |
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Theorem | 6p6e12 9386 |
6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p3e10 9387 |
7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 7p4e11 9388 |
7 + 4 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
AV, 6-Sep-2021.)
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; |
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Theorem | 7p5e12 9389 |
7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p6e13 9390 |
7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p7e14 9391 |
7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p2e10 9392 |
8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) (Revised by Stanislas Polu,
7-Apr-2020.) (Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 8p3e11 9393 |
8 + 3 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
AV, 6-Sep-2021.)
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; |
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Theorem | 8p4e12 9394 |
8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p5e13 9395 |
8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p6e14 9396 |
8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p7e15 9397 |
8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p8e16 9398 |
8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p2e11 9399 |
9 + 2 = 11. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
AV, 6-Sep-2021.)
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; |
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Theorem | 9p3e12 9400 |
9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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