Step |
Hyp |
Ref |
Expression |
1 |
|
decpmulnc.a |
⊢ 𝐴 ∈ ℕ0 |
2 |
|
decpmulnc.b |
⊢ 𝐵 ∈ ℕ0 |
3 |
|
decpmulnc.c |
⊢ 𝐶 ∈ ℕ0 |
4 |
|
decpmulnc.d |
⊢ 𝐷 ∈ ℕ0 |
5 |
|
decpmulnc.1 |
⊢ ( 𝐴 · 𝐶 ) = 𝐸 |
6 |
|
decpmulnc.2 |
⊢ ( ( 𝐴 · 𝐷 ) + ( 𝐵 · 𝐶 ) ) = 𝐹 |
7 |
|
decpmulnc.3 |
⊢ ( 𝐵 · 𝐷 ) = 𝐺 |
8 |
1 2
|
deccl |
⊢ ; 𝐴 𝐵 ∈ ℕ0 |
9 |
|
eqid |
⊢ ; 𝐶 𝐷 = ; 𝐶 𝐷 |
10 |
2 4
|
nn0mulcli |
⊢ ( 𝐵 · 𝐷 ) ∈ ℕ0 |
11 |
7 10
|
eqeltrri |
⊢ 𝐺 ∈ ℕ0 |
12 |
1 4
|
nn0mulcli |
⊢ ( 𝐴 · 𝐷 ) ∈ ℕ0 |
13 |
|
eqid |
⊢ ; 𝐴 𝐵 = ; 𝐴 𝐵 |
14 |
12
|
nn0cni |
⊢ ( 𝐴 · 𝐷 ) ∈ ℂ |
15 |
2 3
|
nn0mulcli |
⊢ ( 𝐵 · 𝐶 ) ∈ ℕ0 |
16 |
15
|
nn0cni |
⊢ ( 𝐵 · 𝐶 ) ∈ ℂ |
17 |
14 16 6
|
addcomli |
⊢ ( ( 𝐵 · 𝐶 ) + ( 𝐴 · 𝐷 ) ) = 𝐹 |
18 |
1 2 12 13 3 5 17
|
decrmanc |
⊢ ( ( ; 𝐴 𝐵 · 𝐶 ) + ( 𝐴 · 𝐷 ) ) = ; 𝐸 𝐹 |
19 |
|
eqid |
⊢ ( 𝐴 · 𝐷 ) = ( 𝐴 · 𝐷 ) |
20 |
4 1 2 13 19 7
|
decmul1 |
⊢ ( ; 𝐴 𝐵 · 𝐷 ) = ; ( 𝐴 · 𝐷 ) 𝐺 |
21 |
8 3 4 9 11 12 18 20
|
decmul2c |
⊢ ( ; 𝐴 𝐵 · ; 𝐶 𝐷 ) = ; ; 𝐸 𝐹 𝐺 |