Description: Calculate a product by long multiplication as a base comparison with other multiplication algorithms.
Conveniently, 7 1 1 has two ones which greatly simplifies calculations like 2 3 5 x. 1 . There isn't a higher level mulcomli saving the lower level uses of mulcomli within 2 3 5 x. 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 are added then this proof would benefit more than ex-decpmul .
For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 or 8t7e56 . (Contributed by Steven Nguyen, 10-Dec-2022) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Assertion | 235t711 | |- ( ; ; 2 3 5 x. ; ; 7 1 1 ) = ; ; ; ; ; 1 6 7 0 8 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 | |- 2 e. NN0 |
|
2 | 3nn0 | |- 3 e. NN0 |
|
3 | 1 2 | deccl | |- ; 2 3 e. NN0 |
4 | 5nn0 | |- 5 e. NN0 |
|
5 | 3 4 | deccl | |- ; ; 2 3 5 e. NN0 |
6 | 7nn0 | |- 7 e. NN0 |
|
7 | 1nn0 | |- 1 e. NN0 |
|
8 | 6 7 | deccl | |- ; 7 1 e. NN0 |
9 | eqid | |- ; ; 7 1 1 = ; ; 7 1 1 |
|
10 | eqid | |- ; 7 1 = ; 7 1 |
|
11 | eqid | |- ; 2 3 = ; 2 3 |
|
12 | 8nn0 | |- 8 e. NN0 |
|
13 | eqid | |- ; ; 2 3 5 = ; ; 2 3 5 |
|
14 | 3 | nn0cni | |- ; 2 3 e. CC |
15 | 2cn | |- 2 e. CC |
|
16 | 3p2e5 | |- ( 3 + 2 ) = 5 |
|
17 | 1 2 1 11 16 | decaddi | |- ( ; 2 3 + 2 ) = ; 2 5 |
18 | 14 15 17 | addcomli | |- ( 2 + ; 2 3 ) = ; 2 5 |
19 | 0nn0 | |- 0 e. NN0 |
|
20 | 4nn0 | |- 4 e. NN0 |
|
21 | 6nn0 | |- 6 e. NN0 |
|
22 | 7 21 | deccl | |- ; 1 6 e. NN0 |
23 | 1 20 | nn0addcli | |- ( 2 + 4 ) e. NN0 |
24 | 7cn | |- 7 e. CC |
|
25 | 7t2e14 | |- ( 7 x. 2 ) = ; 1 4 |
|
26 | 24 15 25 | mulcomli | |- ( 2 x. 7 ) = ; 1 4 |
27 | 4p2e6 | |- ( 4 + 2 ) = 6 |
|
28 | 7 20 1 26 27 | decaddi | |- ( ( 2 x. 7 ) + 2 ) = ; 1 6 |
29 | 3cn | |- 3 e. CC |
|
30 | 7t3e21 | |- ( 7 x. 3 ) = ; 2 1 |
|
31 | 24 29 30 | mulcomli | |- ( 3 x. 7 ) = ; 2 1 |
32 | 6 1 2 11 7 1 28 31 | decmul1c | |- ( ; 2 3 x. 7 ) = ; ; 1 6 1 |
33 | 4cn | |- 4 e. CC |
|
34 | 15 33 | addcli | |- ( 2 + 4 ) e. CC |
35 | ax-1cn | |- 1 e. CC |
|
36 | 33 15 27 | addcomli | |- ( 2 + 4 ) = 6 |
37 | 36 | oveq1i | |- ( ( 2 + 4 ) + 1 ) = ( 6 + 1 ) |
38 | 6p1e7 | |- ( 6 + 1 ) = 7 |
|
39 | 37 38 | eqtri | |- ( ( 2 + 4 ) + 1 ) = 7 |
40 | 34 35 39 | addcomli | |- ( 1 + ( 2 + 4 ) ) = 7 |
41 | 22 7 23 32 40 | decaddi | |- ( ( ; 2 3 x. 7 ) + ( 2 + 4 ) ) = ; ; 1 6 7 |
42 | 5cn | |- 5 e. CC |
|
43 | 7t5e35 | |- ( 7 x. 5 ) = ; 3 5 |
|
44 | 24 42 43 | mulcomli | |- ( 5 x. 7 ) = ; 3 5 |
45 | 3p1e4 | |- ( 3 + 1 ) = 4 |
|
46 | 5p5e10 | |- ( 5 + 5 ) = ; 1 0 |
|
47 | 2 4 4 44 45 46 | decaddci2 | |- ( ( 5 x. 7 ) + 5 ) = ; 4 0 |
48 | 3 4 1 4 13 18 6 19 20 41 47 | decmac | |- ( ( ; ; 2 3 5 x. 7 ) + ( 2 + ; 2 3 ) ) = ; ; ; 1 6 7 0 |
49 | 5 | nn0cni | |- ; ; 2 3 5 e. CC |
50 | 49 | mulid1i | |- ( ; ; 2 3 5 x. 1 ) = ; ; 2 3 5 |
51 | 5p3e8 | |- ( 5 + 3 ) = 8 |
|
52 | 3 4 2 50 51 | decaddi | |- ( ( ; ; 2 3 5 x. 1 ) + 3 ) = ; ; 2 3 8 |
53 | 6 7 1 2 10 11 5 12 3 48 52 | decma2c | |- ( ( ; ; 2 3 5 x. ; 7 1 ) + ; 2 3 ) = ; ; ; ; 1 6 7 0 8 |
54 | 5 8 7 9 4 3 53 50 | decmul2c | |- ( ; ; 2 3 5 x. ; ; 7 1 1 ) = ; ; ; ; ; 1 6 7 0 8 5 |