Metamath Proof Explorer


Theorem 2lgsoddprmlem3d

Description: Lemma 4 for 2lgsoddprmlem3 . (Contributed by AV, 20-Jul-2021)

Ref Expression
Assertion 2lgsoddprmlem3d ( ( ( 7 ↑ 2 ) − 1 ) / 8 ) = ( 2 · 3 )

Proof

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 )