Metamath Proof Explorer

Theorem nummul2c

Description: The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014)

Ref Expression
Hypotheses nummul1c.1 ⊢ 𝑇 ∈ ℕ0
nummul1c.2 ⊢ 𝑃 ∈ ℕ0
nummul1c.3 ⊢ 𝐴 ∈ ℕ0
nummul1c.4 ⊢ 𝐵 ∈ ℕ0
nummul1c.5 ⊢ 𝑁 = ( ( 𝑇 · 𝐴 ) + 𝐵 )
nummul1c.6 ⊢ 𝐷 ∈ ℕ0
nummul1c.7 ⊢ 𝐸 ∈ ℕ0
nummul2c.7 ⊢ ( ( 𝑃 · 𝐴 ) + 𝐸 ) = 𝐶
nummul2c.8 ⊢ ( 𝑃 · 𝐵 ) = ( ( 𝑇 · 𝐸 ) + 𝐷 )
Assertion nummul2c ( 𝑃 · 𝑁 ) = ( ( 𝑇 · 𝐶 ) + 𝐷 )

Proof

Step Hyp Ref Expression
1 nummul1c.1 ⊢ 𝑇 ∈ ℕ0
2 nummul1c.2 ⊢ 𝑃 ∈ ℕ0
3 nummul1c.3 ⊢ 𝐴 ∈ ℕ0
4 nummul1c.4 ⊢ 𝐵 ∈ ℕ0
5 nummul1c.5 ⊢ 𝑁 = ( ( 𝑇 · 𝐴 ) + 𝐵 )
6 nummul1c.6 ⊢ 𝐷 ∈ ℕ0
7 nummul1c.7 ⊢ 𝐸 ∈ ℕ0
8 nummul2c.7 ⊢ ( ( 𝑃 · 𝐴 ) + 𝐸 ) = 𝐶
9 nummul2c.8 ⊢ ( 𝑃 · 𝐵 ) = ( ( 𝑇 · 𝐸 ) + 𝐷 )
10 1 3 4 numcl ⊢ ( ( 𝑇 · 𝐴 ) + 𝐵 ) ∈ ℕ0
11 5 10 eqeltri ⊢ 𝑁 ∈ ℕ0
12 11 nn0cni ⊢ 𝑁 ∈ ℂ
13 2 nn0cni ⊢ 𝑃 ∈ ℂ
14 3 nn0cni ⊢ 𝐴 ∈ ℂ
15 14 13 mulcomi ⊢ ( 𝐴 · 𝑃 ) = ( 𝑃 · 𝐴 )
16 15 oveq1i ⊢ ( ( 𝐴 · 𝑃 ) + 𝐸 ) = ( ( 𝑃 · 𝐴 ) + 𝐸 )
17 16 8 eqtri ⊢ ( ( 𝐴 · 𝑃 ) + 𝐸 ) = 𝐶
18 4 nn0cni ⊢ 𝐵 ∈ ℂ
19 13 18 9 mulcomli ⊢ ( 𝐵 · 𝑃 ) = ( ( 𝑇 · 𝐸 ) + 𝐷 )
20 1 2 3 4 5 6 7 17 19 nummul1c ⊢ ( 𝑁 · 𝑃 ) = ( ( 𝑇 · 𝐶 ) + 𝐷 )
21 12 13 20 mulcomli ⊢ ( 𝑃 · 𝑁 ) = ( ( 𝑇 · 𝐶 ) + 𝐷 )