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 )