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 = 2187

Proof

Step Hyp Ref Expression
1 3nn0 3 0
2 6nn0 6 0
3 6p1e7 6 + 1 = 7
4 7nn0 7 0
5 2nn0 2 0
6 4 5 deccl 72 0
7 9nn0 9 0
8 3cn 3
9 2cn 2
10 3t2e6 3 2 = 6
11 8 9 10 mulcomli 2 3 = 6
12 3exp3 3 3 = 27
13 5 4 deccl 27 0
14 eqid 27 = 27
15 1nn0 1 0
16 8nn0 8 0
17 15 16 deccl 18 0
18 0nn0 0 0
19 5 dec0h 2 = 02
20 eqid 18 = 18
21 13 nn0cni 27
22 21 mul02i 0 27 = 0
23 6cn 6
24 ax-1cn 1
25 23 24 3 addcomli 1 + 6 = 7
26 22 25 oveq12i 0 27 + 1 + 6 = 0 + 7
27 7cn 7
28 27 addid2i 0 + 7 = 7
29 26 28 eqtri 0 27 + 1 + 6 = 7
30 16 dec0h 8 = 08
31 2t2e4 2 2 = 4
32 9 addid2i 0 + 2 = 2
33 31 32 oveq12i 2 2 + 0 + 2 = 4 + 2
34 4p2e6 4 + 2 = 6
35 33 34 eqtri 2 2 + 0 + 2 = 6
36 4nn0 4 0
37 7t2e14 7 2 = 14
38 27 9 37 mulcomli 2 7 = 14
39 1p1e2 1 + 1 = 2
40 8cn 8
41 4cn 4
42 8p4e12 8 + 4 = 12
43 40 41 42 addcomli 4 + 8 = 12
44 15 36 16 38 39 5 43 decaddci 2 7 + 8 = 22
45 5 4 18 16 14 30 5 5 5 35 44 decma2c 2 27 + 8 = 62
46 18 5 15 16 19 20 13 5 2 29 45 decmac 2 27 + 18 = 72
47 4p4e8 4 + 4 = 8
48 15 36 36 37 47 decaddi 7 2 + 4 = 18
49 7t7e49 7 7 = 49
50 4 5 4 14 7 36 48 49 decmul2c 7 27 = 189
51 13 5 4 14 7 17 46 50 decmul1c 27 27 = 729
52 1 1 11 12 51 numexp2x 3 6 = 729
53 eqid 72 = 72
54 7t3e21 7 3 = 21
55 1p0e1 1 + 0 = 1
56 5 15 18 54 55 decaddi 7 3 + 0 = 21
57 11 oveq1i 2 3 + 2 = 6 + 2
58 6p2e8 6 + 2 = 8
59 57 58 eqtri 2 3 + 2 = 8
60 4 5 18 5 53 19 1 56 59 decma 72 3 + 2 = 218
61 9t3e27 9 3 = 27
62 1 6 7 52 4 5 60 61 decmul1c 3 6 3 = 2187
63 1 2 3 62 numexpp1 3 7 = 2187