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	$$⊢A\in {\mathbb{N}}_0$$
	decma.b	$$⊢B\in {\mathbb{N}}_0$$
	decma.c	$$⊢C\in {\mathbb{N}}_0$$
	decma.d	$$⊢D\in {\mathbb{N}}_0$$
	decma.m	No typesetting found for |- M = ; A B with typecode |-
	decma.n	No typesetting found for |- N = ; C D with typecode |-
	decma.p	$$⊢P\in {\mathbb{N}}_0$$
	decma.e	$$⊢AP+C=E$$
	decma.f	$$⊢BP+D=F$$
Assertion	decma	Could not format assertion : No typesetting found for |- ( ( M x. P ) + N ) = ; E F with typecode |-

Proof

Step	Hyp	Ref	Expression
1		decma.a	$$⊢A\in {\mathbb{N}}_0$$
2		decma.b	$$⊢B\in {\mathbb{N}}_0$$
3		decma.c	$$⊢C\in {\mathbb{N}}_0$$
4		decma.d	$$⊢D\in {\mathbb{N}}_0$$
5		decma.m	Could not format M = ; A B : No typesetting found for |- M = ; A B with typecode |-
6		decma.n	Could not format N = ; C D : No typesetting found for |- N = ; C D with typecode |-
7		decma.p	$$⊢P\in {\mathbb{N}}_0$$
8		decma.e	$$⊢AP+C=E$$
9		decma.f	$$⊢BP+D=F$$
10		10nn0	$$⊢10\in {\mathbb{N}}_0$$
11		dfdec10	Could not format ; A B = ( ( ; 1 0 x. A ) + B ) : No typesetting found for |- ; A B = ( ( ; 1 0 x. A ) + B ) with typecode |-
12	5 11	eqtri	$$⊢M=10A+B$$
13		dfdec10	Could not format ; C D = ( ( ; 1 0 x. C ) + D ) : No typesetting found for |- ; C D = ( ( ; 1 0 x. C ) + D ) with typecode |-
14	6 13	eqtri	$$⊢N=10C+D$$
15	10 1 2 3 4 12 14 7 8 9	numma	$$⊢MP+N=10E+F$$
16		dfdec10	Could not format ; E F = ( ( ; 1 0 x. E ) + F ) : No typesetting found for |- ; E F = ( ( ; 1 0 x. E ) + F ) with typecode |-
17	15 16	eqtr4i	Could not format ( ( M x. P ) + N ) = ; E F : No typesetting found for |- ( ( M x. P ) + N ) = ; E F with typecode |-