Metamath Proof Explorer
Description: The successor of a decimal integer (with carry). (Contributed by Mario
Carneiro, 18-Feb-2014) (Revised by AV, 6-Sep-2021)
|
|
Ref |
Expression |
|
Hypotheses |
decsucc.1 |
⊢ 𝐴 ∈ ℕ0 |
|
|
decsucc.2 |
⊢ ( 𝐴 + 1 ) = 𝐵 |
|
|
decsucc.3 |
⊢ 𝑁 = ; 𝐴 9 |
|
Assertion |
decsucc |
⊢ ( 𝑁 + 1 ) = ; 𝐵 0 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
decsucc.1 |
⊢ 𝐴 ∈ ℕ0 |
2 |
|
decsucc.2 |
⊢ ( 𝐴 + 1 ) = 𝐵 |
3 |
|
decsucc.3 |
⊢ 𝑁 = ; 𝐴 9 |
4 |
|
9nn0 |
⊢ 9 ∈ ℕ0 |
5 |
|
9p1e10 |
⊢ ( 9 + 1 ) = ; 1 0 |
6 |
5
|
eqcomi |
⊢ ; 1 0 = ( 9 + 1 ) |
7 |
|
dfdec10 |
⊢ ; 𝐴 9 = ( ( ; 1 0 · 𝐴 ) + 9 ) |
8 |
3 7
|
eqtri |
⊢ 𝑁 = ( ( ; 1 0 · 𝐴 ) + 9 ) |
9 |
4 6 1 2 8
|
numsucc |
⊢ ( 𝑁 + 1 ) = ( ( ; 1 0 · 𝐵 ) + 0 ) |
10 |
|
dfdec10 |
⊢ ; 𝐵 0 = ( ( ; 1 0 · 𝐵 ) + 0 ) |
11 |
9 10
|
eqtr4i |
⊢ ( 𝑁 + 1 ) = ; 𝐵 0 |