Metamath Proof Explorer
Description: The product of a numeral with a number (no carry). (Contributed by AV, 15-Jun-2021)
|
|
Ref |
Expression |
|
Hypotheses |
decmulnc.n |
⊢ 𝑁 ∈ ℕ0 |
|
|
decmulnc.a |
⊢ 𝐴 ∈ ℕ0 |
|
|
decmulnc.b |
⊢ 𝐵 ∈ ℕ0 |
|
Assertion |
decmulnc |
⊢ ( 𝑁 · ; 𝐴 𝐵 ) = ; ( 𝑁 · 𝐴 ) ( 𝑁 · 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
decmulnc.n |
⊢ 𝑁 ∈ ℕ0 |
2 |
|
decmulnc.a |
⊢ 𝐴 ∈ ℕ0 |
3 |
|
decmulnc.b |
⊢ 𝐵 ∈ ℕ0 |
4 |
|
eqid |
⊢ ; 𝐴 𝐵 = ; 𝐴 𝐵 |
5 |
1 3
|
nn0mulcli |
⊢ ( 𝑁 · 𝐵 ) ∈ ℕ0 |
6 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
7 |
1 2
|
nn0mulcli |
⊢ ( 𝑁 · 𝐴 ) ∈ ℕ0 |
8 |
7
|
nn0cni |
⊢ ( 𝑁 · 𝐴 ) ∈ ℂ |
9 |
8
|
addid1i |
⊢ ( ( 𝑁 · 𝐴 ) + 0 ) = ( 𝑁 · 𝐴 ) |
10 |
5
|
dec0h |
⊢ ( 𝑁 · 𝐵 ) = ; 0 ( 𝑁 · 𝐵 ) |
11 |
1 2 3 4 5 6 9 10
|
decmul2c |
⊢ ( 𝑁 · ; 𝐴 𝐵 ) = ; ( 𝑁 · 𝐴 ) ( 𝑁 · 𝐵 ) |