Description: The lcm of 420 and 8 is 840. (Contributed by metakunt, 25-Apr-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | 420lcm8e840 | |- ( ; ; 4 2 0 lcm 8 ) = ; ; 8 4 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 | |- 4 e. NN0 |
|
2 | 2nn | |- 2 e. NN |
|
3 | 1 2 | decnncl | |- ; 4 2 e. NN |
4 | 3 | decnncl2 | |- ; ; 4 2 0 e. NN |
5 | 8nn | |- 8 e. NN |
|
6 | 4nn | |- 4 e. NN |
|
7 | 8nn0 | |- 8 e. NN0 |
|
8 | 7 6 | decnncl | |- ; 8 4 e. NN |
9 | 8 | decnncl2 | |- ; ; 8 4 0 e. NN |
10 | 420gcd8e4 | |- ( ; ; 4 2 0 gcd 8 ) = 4 |
|
11 | eqid | |- ( 4 x. ; ; 8 4 0 ) = ( 4 x. ; ; 8 4 0 ) |
|
12 | 4 5 | mulcomnni | |- ( ; ; 4 2 0 x. 8 ) = ( 8 x. ; ; 4 2 0 ) |
13 | 4t2e8 | |- ( 4 x. 2 ) = 8 |
|
14 | 13 | oveq1i | |- ( ( 4 x. 2 ) x. ; ; 4 2 0 ) = ( 8 x. ; ; 4 2 0 ) |
15 | 12 14 | eqtr4i | |- ( ; ; 4 2 0 x. 8 ) = ( ( 4 x. 2 ) x. ; ; 4 2 0 ) |
16 | 6 2 4 | mulassnni | |- ( ( 4 x. 2 ) x. ; ; 4 2 0 ) = ( 4 x. ( 2 x. ; ; 4 2 0 ) ) |
17 | 15 16 | eqtri | |- ( ; ; 4 2 0 x. 8 ) = ( 4 x. ( 2 x. ; ; 4 2 0 ) ) |
18 | 2 | nnnn0i | |- 2 e. NN0 |
19 | 3 | nnnn0i | |- ; 4 2 e. NN0 |
20 | 0nn0 | |- 0 e. NN0 |
|
21 | eqid | |- ; ; 4 2 0 = ; ; 4 2 0 |
|
22 | eqid | |- ; 4 2 = ; 4 2 |
|
23 | 4cn | |- 4 e. CC |
|
24 | 2cn | |- 2 e. CC |
|
25 | 23 24 13 | mulcomli | |- ( 2 x. 4 ) = 8 |
26 | 25 | oveq1i | |- ( ( 2 x. 4 ) + 0 ) = ( 8 + 0 ) |
27 | 8cn | |- 8 e. CC |
|
28 | 27 | addid1i | |- ( 8 + 0 ) = 8 |
29 | 26 28 | eqtri | |- ( ( 2 x. 4 ) + 0 ) = 8 |
30 | 2t2e4 | |- ( 2 x. 2 ) = 4 |
|
31 | 1 | dec0h | |- 4 = ; 0 4 |
32 | 31 | eqcomi | |- ; 0 4 = 4 |
33 | 30 32 | eqtr4i | |- ( 2 x. 2 ) = ; 0 4 |
34 | 18 1 18 22 1 20 29 33 | decmul2c | |- ( 2 x. ; 4 2 ) = ; 8 4 |
35 | 23 | addid1i | |- ( 4 + 0 ) = 4 |
36 | 7 1 20 34 35 | decaddi | |- ( ( 2 x. ; 4 2 ) + 0 ) = ; 8 4 |
37 | 2t0e0 | |- ( 2 x. 0 ) = 0 |
|
38 | 20 | dec0h | |- 0 = ; 0 0 |
39 | 38 | eqcomi | |- ; 0 0 = 0 |
40 | 37 39 | eqtr4i | |- ( 2 x. 0 ) = ; 0 0 |
41 | 18 19 20 21 20 20 36 40 | decmul2c | |- ( 2 x. ; ; 4 2 0 ) = ; ; 8 4 0 |
42 | 41 | oveq2i | |- ( 4 x. ( 2 x. ; ; 4 2 0 ) ) = ( 4 x. ; ; 8 4 0 ) |
43 | 17 42 | eqtri | |- ( ; ; 4 2 0 x. 8 ) = ( 4 x. ; ; 8 4 0 ) |
44 | 4 5 6 9 10 11 43 | lcmeprodgcdi | |- ( ; ; 4 2 0 lcm 8 ) = ; ; 8 4 0 |