Metamath Proof Explorer

Theorem 3exp7

Description: 3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024)

Ref Expression
Assertion 3exp7 
|- ( 3 ^ 7 ) = ; ; ; 2 1 8 7

Proof

Step Hyp Ref Expression
1 3nn0 
 |-  3 e. NN0
2 6nn0 
 |-  6 e. NN0
3 6p1e7 
 |-  ( 6 + 1 ) = 7
4 7nn0 
 |-  7 e. NN0
5 2nn0 
 |-  2 e. NN0
6 4 5 deccl 
 |-  ; 7 2 e. NN0
7 9nn0 
 |-  9 e. NN0
8 2t3e6 
 |-  ( 2 x. 3 ) = 6
9 3exp3 
 |-  ( 3 ^ 3 ) = ; 2 7
10 5 4 deccl 
 |-  ; 2 7 e. NN0
11 eqid 
 |-  ; 2 7 = ; 2 7
12 1nn0 
 |-  1 e. NN0
13 8nn0 
 |-  8 e. NN0
14 12 13 deccl 
 |-  ; 1 8 e. NN0
15 0nn0 
 |-  0 e. NN0
16 5 dec0h 
 |-  2 = ; 0 2
17 eqid 
 |-  ; 1 8 = ; 1 8
18 10 nn0cni 
 |-  ; 2 7 e. CC
19 18 mul02i 
 |-  ( 0 x. ; 2 7 ) = 0
20 6cn 
 |-  6 e. CC
21 ax-1cn 
 |-  1 e. CC
22 20 21 3 addcomli 
 |-  ( 1 + 6 ) = 7
23 19 22 oveq12i 
 |-  ( ( 0 x. ; 2 7 ) + ( 1 + 6 ) ) = ( 0 + 7 )
24 7cn 
 |-  7 e. CC
25 24 addlidi 
 |-  ( 0 + 7 ) = 7
26 23 25 eqtri 
 |-  ( ( 0 x. ; 2 7 ) + ( 1 + 6 ) ) = 7
27 13 dec0h 
 |-  8 = ; 0 8
28 2t2e4 
 |-  ( 2 x. 2 ) = 4
29 2cn 
 |-  2 e. CC
30 29 addlidi 
 |-  ( 0 + 2 ) = 2
31 28 30 oveq12i 
 |-  ( ( 2 x. 2 ) + ( 0 + 2 ) ) = ( 4 + 2 )
32 4p2e6 
 |-  ( 4 + 2 ) = 6
33 31 32 eqtri 
 |-  ( ( 2 x. 2 ) + ( 0 + 2 ) ) = 6
34 4nn0 
 |-  4 e. NN0
35 7t2e14 
 |-  ( 7 x. 2 ) = ; 1 4
36 24 29 35 mulcomli 
 |-  ( 2 x. 7 ) = ; 1 4
37 1p1e2 
 |-  ( 1 + 1 ) = 2
38 8cn 
 |-  8 e. CC
39 4cn 
 |-  4 e. CC
40 8p4e12 
 |-  ( 8 + 4 ) = ; 1 2
41 38 39 40 addcomli 
 |-  ( 4 + 8 ) = ; 1 2
42 12 34 13 36 37 5 41 decaddci 
 |-  ( ( 2 x. 7 ) + 8 ) = ; 2 2
43 5 4 15 13 11 27 5 5 5 33 42 decma2c 
 |-  ( ( 2 x. ; 2 7 ) + 8 ) = ; 6 2
44 15 5 12 13 16 17 10 5 2 26 43 decmac 
 |-  ( ( 2 x. ; 2 7 ) + ; 1 8 ) = ; 7 2
45 4p4e8 
 |-  ( 4 + 4 ) = 8
46 12 34 34 35 45 decaddi 
 |-  ( ( 7 x. 2 ) + 4 ) = ; 1 8
47 7t7e49 
 |-  ( 7 x. 7 ) = ; 4 9
48 4 5 4 11 7 34 46 47 decmul2c 
 |-  ( 7 x. ; 2 7 ) = ; ; 1 8 9
49 10 5 4 11 7 14 44 48 decmul1c 
 |-  ( ; 2 7 x. ; 2 7 ) = ; ; 7 2 9
50 1 1 8 9 49 numexp2x 
 |-  ( 3 ^ 6 ) = ; ; 7 2 9
51 eqid 
 |-  ; 7 2 = ; 7 2
52 7t3e21 
 |-  ( 7 x. 3 ) = ; 2 1
53 1p0e1 
 |-  ( 1 + 0 ) = 1
54 5 12 15 52 53 decaddi 
 |-  ( ( 7 x. 3 ) + 0 ) = ; 2 1
55 8 oveq1i 
 |-  ( ( 2 x. 3 ) + 2 ) = ( 6 + 2 )
56 6p2e8 
 |-  ( 6 + 2 ) = 8
57 55 56 eqtri 
 |-  ( ( 2 x. 3 ) + 2 ) = 8
58 4 5 15 5 51 16 1 54 57 decma 
 |-  ( ( ; 7 2 x. 3 ) + 2 ) = ; ; 2 1 8
59 9t3e27 
 |-  ( 9 x. 3 ) = ; 2 7
60 1 6 7 50 4 5 58 59 decmul1c 
 |-  ( ( 3 ^ 6 ) x. 3 ) = ; ; ; 2 1 8 7
61 1 2 3 60 numexpp1 
 |-  ( 3 ^ 7 ) = ; ; ; 2 1 8 7