Metamath Proof Explorer


Theorem 420lcm8e840

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

Proof

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