Step |
Hyp |
Ref |
Expression |
1 |
|
decma.a |
⊢ 𝐴 ∈ ℕ0 |
2 |
|
decma.b |
⊢ 𝐵 ∈ ℕ0 |
3 |
|
decma.c |
⊢ 𝐶 ∈ ℕ0 |
4 |
|
decma.d |
⊢ 𝐷 ∈ ℕ0 |
5 |
|
decma.m |
⊢ 𝑀 = ; 𝐴 𝐵 |
6 |
|
decma.n |
⊢ 𝑁 = ; 𝐶 𝐷 |
7 |
|
decaddc.e |
⊢ ( ( 𝐴 + 𝐶 ) + 1 ) = 𝐸 |
8 |
|
decaddc.f |
⊢ 𝐹 ∈ ℕ0 |
9 |
|
decaddc.2 |
⊢ ( 𝐵 + 𝐷 ) = ; 1 𝐹 |
10 |
|
10nn0 |
⊢ ; 1 0 ∈ ℕ0 |
11 |
|
dfdec10 |
⊢ ; 𝐴 𝐵 = ( ( ; 1 0 · 𝐴 ) + 𝐵 ) |
12 |
5 11
|
eqtri |
⊢ 𝑀 = ( ( ; 1 0 · 𝐴 ) + 𝐵 ) |
13 |
|
dfdec10 |
⊢ ; 𝐶 𝐷 = ( ( ; 1 0 · 𝐶 ) + 𝐷 ) |
14 |
6 13
|
eqtri |
⊢ 𝑁 = ( ( ; 1 0 · 𝐶 ) + 𝐷 ) |
15 |
|
dfdec10 |
⊢ ; 1 𝐹 = ( ( ; 1 0 · 1 ) + 𝐹 ) |
16 |
9 15
|
eqtri |
⊢ ( 𝐵 + 𝐷 ) = ( ( ; 1 0 · 1 ) + 𝐹 ) |
17 |
10 1 2 3 4 12 14 8 7 16
|
numaddc |
⊢ ( 𝑀 + 𝑁 ) = ( ( ; 1 0 · 𝐸 ) + 𝐹 ) |
18 |
|
dfdec10 |
⊢ ; 𝐸 𝐹 = ( ( ; 1 0 · 𝐸 ) + 𝐹 ) |
19 |
17 18
|
eqtr4i |
⊢ ( 𝑀 + 𝑁 ) = ; 𝐸 𝐹 |