Step |
Hyp |
Ref |
Expression |
1 |
|
6cn |
⊢ 6 ∈ ℂ |
2 |
|
8cn |
⊢ 8 ∈ ℂ |
3 |
|
0re |
⊢ 0 ∈ ℝ |
4 |
|
8pos |
⊢ 0 < 8 |
5 |
3 4
|
gtneii |
⊢ 8 ≠ 0 |
6 |
1 2 5
|
divcan4i |
⊢ ( ( 6 · 8 ) / 8 ) = 6 |
7 |
1 2
|
mulcli |
⊢ ( 6 · 8 ) ∈ ℂ |
8 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
9 |
|
4p3e7 |
⊢ ( 4 + 3 ) = 7 |
10 |
9
|
eqcomi |
⊢ 7 = ( 4 + 3 ) |
11 |
10
|
oveq1i |
⊢ ( 7 ↑ 2 ) = ( ( 4 + 3 ) ↑ 2 ) |
12 |
|
4cn |
⊢ 4 ∈ ℂ |
13 |
|
3cn |
⊢ 3 ∈ ℂ |
14 |
12 13
|
binom2i |
⊢ ( ( 4 + 3 ) ↑ 2 ) = ( ( ( 4 ↑ 2 ) + ( 2 · ( 4 · 3 ) ) ) + ( 3 ↑ 2 ) ) |
15 |
|
sq4e2t8 |
⊢ ( 4 ↑ 2 ) = ( 2 · 8 ) |
16 |
|
2cn |
⊢ 2 ∈ ℂ |
17 |
|
4t2e8 |
⊢ ( 4 · 2 ) = 8 |
18 |
12 16 17
|
mulcomli |
⊢ ( 2 · 4 ) = 8 |
19 |
18
|
oveq1i |
⊢ ( ( 2 · 4 ) · 3 ) = ( 8 · 3 ) |
20 |
16 12 13
|
mulassi |
⊢ ( ( 2 · 4 ) · 3 ) = ( 2 · ( 4 · 3 ) ) |
21 |
2 13
|
mulcomi |
⊢ ( 8 · 3 ) = ( 3 · 8 ) |
22 |
19 20 21
|
3eqtr3i |
⊢ ( 2 · ( 4 · 3 ) ) = ( 3 · 8 ) |
23 |
15 22
|
oveq12i |
⊢ ( ( 4 ↑ 2 ) + ( 2 · ( 4 · 3 ) ) ) = ( ( 2 · 8 ) + ( 3 · 8 ) ) |
24 |
16 13 2
|
adddiri |
⊢ ( ( 2 + 3 ) · 8 ) = ( ( 2 · 8 ) + ( 3 · 8 ) ) |
25 |
|
3p2e5 |
⊢ ( 3 + 2 ) = 5 |
26 |
13 16 25
|
addcomli |
⊢ ( 2 + 3 ) = 5 |
27 |
26
|
oveq1i |
⊢ ( ( 2 + 3 ) · 8 ) = ( 5 · 8 ) |
28 |
23 24 27
|
3eqtr2i |
⊢ ( ( 4 ↑ 2 ) + ( 2 · ( 4 · 3 ) ) ) = ( 5 · 8 ) |
29 |
|
sq3 |
⊢ ( 3 ↑ 2 ) = 9 |
30 |
|
df-9 |
⊢ 9 = ( 8 + 1 ) |
31 |
29 30
|
eqtri |
⊢ ( 3 ↑ 2 ) = ( 8 + 1 ) |
32 |
28 31
|
oveq12i |
⊢ ( ( ( 4 ↑ 2 ) + ( 2 · ( 4 · 3 ) ) ) + ( 3 ↑ 2 ) ) = ( ( 5 · 8 ) + ( 8 + 1 ) ) |
33 |
|
5cn |
⊢ 5 ∈ ℂ |
34 |
33 2
|
mulcli |
⊢ ( 5 · 8 ) ∈ ℂ |
35 |
34 2 8
|
addassi |
⊢ ( ( ( 5 · 8 ) + 8 ) + 1 ) = ( ( 5 · 8 ) + ( 8 + 1 ) ) |
36 |
|
df-6 |
⊢ 6 = ( 5 + 1 ) |
37 |
36
|
oveq1i |
⊢ ( 6 · 8 ) = ( ( 5 + 1 ) · 8 ) |
38 |
33
|
a1i |
⊢ ( 8 ∈ ℂ → 5 ∈ ℂ ) |
39 |
|
id |
⊢ ( 8 ∈ ℂ → 8 ∈ ℂ ) |
40 |
38 39
|
adddirp1d |
⊢ ( 8 ∈ ℂ → ( ( 5 + 1 ) · 8 ) = ( ( 5 · 8 ) + 8 ) ) |
41 |
2 40
|
ax-mp |
⊢ ( ( 5 + 1 ) · 8 ) = ( ( 5 · 8 ) + 8 ) |
42 |
37 41
|
eqtri |
⊢ ( 6 · 8 ) = ( ( 5 · 8 ) + 8 ) |
43 |
42
|
eqcomi |
⊢ ( ( 5 · 8 ) + 8 ) = ( 6 · 8 ) |
44 |
43
|
oveq1i |
⊢ ( ( ( 5 · 8 ) + 8 ) + 1 ) = ( ( 6 · 8 ) + 1 ) |
45 |
32 35 44
|
3eqtr2i |
⊢ ( ( ( 4 ↑ 2 ) + ( 2 · ( 4 · 3 ) ) ) + ( 3 ↑ 2 ) ) = ( ( 6 · 8 ) + 1 ) |
46 |
14 45
|
eqtri |
⊢ ( ( 4 + 3 ) ↑ 2 ) = ( ( 6 · 8 ) + 1 ) |
47 |
11 46
|
eqtri |
⊢ ( 7 ↑ 2 ) = ( ( 6 · 8 ) + 1 ) |
48 |
7 8 47
|
mvrraddi |
⊢ ( ( 7 ↑ 2 ) − 1 ) = ( 6 · 8 ) |
49 |
48
|
oveq1i |
⊢ ( ( ( 7 ↑ 2 ) − 1 ) / 8 ) = ( ( 6 · 8 ) / 8 ) |
50 |
|
3t2e6 |
⊢ ( 3 · 2 ) = 6 |
51 |
13 16 50
|
mulcomli |
⊢ ( 2 · 3 ) = 6 |
52 |
6 49 51
|
3eqtr4i |
⊢ ( ( ( 7 ↑ 2 ) − 1 ) / 8 ) = ( 2 · 3 ) |