Theorem decma

Description: Perform a multiply-add of two numerals M and N against a fixed multiplicand P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014) (Revised by AV, 6-Sep-2021)

	Ref	Expression
Hypotheses	decma.a	⊢ 𝐴 ∈ ℕ0
	decma.b	⊢ 𝐵 ∈ ℕ0
	decma.c	⊢ 𝐶 ∈ ℕ0
	decma.d	⊢ 𝐷 ∈ ℕ0
	decma.m	⊢ 𝑀 = ; 𝐴 𝐵
	decma.n	⊢ 𝑁 = ; 𝐶 𝐷
	decma.p	⊢ 𝑃 ∈ ℕ0
	decma.e	⊢ ( ( 𝐴 · 𝑃 ) + 𝐶 ) = 𝐸
	decma.f	⊢ ( ( 𝐵 · 𝑃 ) + 𝐷 ) = 𝐹
Assertion	decma	⊢ ( ( 𝑀 · 𝑃 ) + 𝑁 ) = ; 𝐸 𝐹

Proof

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		decma.p	⊢ 𝑃 ∈ ℕ0
8		decma.e	⊢ ( ( 𝐴 · 𝑃 ) + 𝐶 ) = 𝐸
9		decma.f	⊢ ( ( 𝐵 · 𝑃 ) + 𝐷 ) = 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	10 1 2 3 4 12 14 7 8 9	numma	⊢ ( ( 𝑀 · 𝑃 ) + 𝑁 ) = ( ( ; 1 0 · 𝐸 ) + 𝐹 )
16		dfdec10	⊢ ; 𝐸 𝐹 = ( ( ; 1 0 · 𝐸 ) + 𝐹 )
17	15 16	eqtr4i	⊢ ( ( 𝑀 · 𝑃 ) + 𝑁 ) = ; 𝐸 𝐹