Theorem List for Intuitionistic Logic Explorer - 9201-9300 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | decnncl 9201 |
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 9202 |
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 9203 |
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 9204 |
Closure for a decimal integer (zero units place). (Contributed by Mario
Carneiro, 9-Mar-2015.)
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Theorem | decnncl2 9205 |
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 9206 |
Comparing two decimal integers (equal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | numltc 9207 |
Comparing two decimal integers (unequal higher places). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | le9lt10 9208 |
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 9209 |
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 9210 |
Comparing two decimal integers (unequal higher places). (Contributed
by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ; ; |
|
Theorem | declth 9211 |
Comparing two decimal integers (unequal higher places). (Contributed
by AV, 8-Sep-2021.)
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; ; |
|
Theorem | decsuc 9212 |
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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; ; |
|
Theorem | 3declth 9213 |
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 9214 |
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 9215 |
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 9216 |
Comparing two decimal integers (unequal higher places). (Contributed by
AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.)
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; ; |
|
Theorem | declei 9217 |
Comparing a digit to a decimal integer. (Contributed by AV,
17-Aug-2021.)
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; |
|
Theorem | numlti 9218 |
Comparing a digit to a decimal integer. (Contributed by Mario Carneiro,
18-Feb-2014.)
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Theorem | declti 9219 |
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 9220 |
Comparing a digit to a decimal integer. (Contributed by AV,
8-Sep-2021.)
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; |
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Theorem | numsucc 9221 |
The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014.)
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|
Theorem | decsucc 9222 |
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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; ; |
|
Theorem | 1e0p1 9223 |
The successor of zero. (Contributed by Mario Carneiro, 18-Feb-2014.)
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Theorem | dec10p 9224 |
Ten plus an integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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;
; |
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Theorem | numma 9225 |
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 9226 |
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 9227 |
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 9228 |
Add two decimal integers and (no
carry). (Contributed by
Mario Carneiro, 18-Feb-2014.)
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Theorem | numaddc 9229 |
Add two decimal integers and (with
carry). (Contributed
by Mario Carneiro, 18-Feb-2014.)
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Theorem | nummul1c 9230 |
The product of a decimal integer with a number. (Contributed by Mario
Carneiro, 18-Feb-2014.)
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Theorem | nummul2c 9231 |
The product of a decimal integer with a number (with carry).
(Contributed by Mario Carneiro, 18-Feb-2014.)
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Theorem | decma 9232 |
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 9233 |
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 9234 |
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 9235 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
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; ;
; |
|
Theorem | decaddc 9236 |
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 | decaddc2 9237 |
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 9238 |
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 9239 |
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 9240 |
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 9241 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
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;
; |
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Theorem | decaddci 9242 |
Add two numerals and
(no carry).
(Contributed by Mario
Carneiro, 18-Feb-2014.)
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;
;
; |
|
Theorem | decaddci2 9243 |
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 9244 |
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 9245 |
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 9246 |
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 9247 |
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 9248 |
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 9249 |
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 9250 |
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 9251 |
Lemma for 6p5e11 9254 and related theorems. (Contributed by Mario
Carneiro, 19-Apr-2015.)
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;
; |
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Theorem | 5p5e10 9252 |
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 9253 |
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 9254 |
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 9255 |
6 + 6 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p3e10 9256 |
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 9257 |
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 9258 |
7 + 5 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p6e13 9259 |
7 + 6 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7p7e14 9260 |
7 + 7 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p2e10 9261 |
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 9262 |
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 9263 |
8 + 4 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p5e13 9264 |
8 + 5 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p6e14 9265 |
8 + 6 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p7e15 9266 |
8 + 7 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8p8e16 9267 |
8 + 8 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p2e11 9268 |
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 9269 |
9 + 3 = 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p4e13 9270 |
9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p5e14 9271 |
9 + 5 = 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p6e15 9272 |
9 + 6 = 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p7e16 9273 |
9 + 7 = 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p8e17 9274 |
9 + 8 = 17. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 9p9e18 9275 |
9 + 9 = 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 10p10e20 9276 |
10 + 10 = 20. (Contributed by Mario Carneiro, 19-Apr-2015.) (Revised by
AV, 6-Sep-2021.)
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; ; ; |
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Theorem | 10m1e9 9277 |
10 - 1 = 9. (Contributed by AV, 6-Sep-2021.)
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; |
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Theorem | 4t3lem 9278 |
Lemma for 4t3e12 9279 and related theorems. (Contributed by Mario
Carneiro, 19-Apr-2015.)
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Theorem | 4t3e12 9279 |
4 times 3 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 4t4e16 9280 |
4 times 4 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 5t2e10 9281 |
5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) (Revised by AV,
4-Sep-2021.)
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; |
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Theorem | 5t3e15 9282 |
5 times 3 equals 15. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 5t4e20 9283 |
5 times 4 equals 20. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 5t5e25 9284 |
5 times 5 equals 25. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 6t2e12 9285 |
6 times 2 equals 12. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 6t3e18 9286 |
6 times 3 equals 18. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 6t4e24 9287 |
6 times 4 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 6t5e30 9288 |
6 times 5 equals 30. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 6t6e36 9289 |
6 times 6 equals 36. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 7t2e14 9290 |
7 times 2 equals 14. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7t3e21 9291 |
7 times 3 equals 21. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7t4e28 9292 |
7 times 4 equals 28. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7t5e35 9293 |
7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7t6e42 9294 |
7 times 6 equals 42. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 7t7e49 9295 |
7 times 7 equals 49. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8t2e16 9296 |
8 times 2 equals 16. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8t3e24 9297 |
8 times 3 equals 24. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8t4e32 9298 |
8 times 4 equals 32. (Contributed by Mario Carneiro, 19-Apr-2015.)
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; |
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Theorem | 8t5e40 9299 |
8 times 5 equals 40. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |
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Theorem | 8t6e48 9300 |
8 times 6 equals 48. (Contributed by Mario Carneiro, 19-Apr-2015.)
(Revised by AV, 6-Sep-2021.)
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; |