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 e. NN0
	decma.b	|- B e. NN0
	decma.c	|- C e. NN0
	decma.d	|- D e. NN0
	decma.m	|- M = ; A B
	decma.n	|- N = ; C D
	decma.p	|- P e. NN0
	decma.e	|- ( ( A x. P ) + C ) = E
	decma.f	|- ( ( B x. P ) + D ) = F
Assertion	decma	|- ( ( M x. P ) + N ) = ; E F

Proof

Step	Hyp	Ref	Expression
1		decma.a	|- A e. NN0
2		decma.b	|- B e. NN0
3		decma.c	|- C e. NN0
4		decma.d	|- D e. NN0
5		decma.m	|- M = ; A B
6		decma.n	|- N = ; C D
7		decma.p	|- P e. NN0
8		decma.e	|- ( ( A x. P ) + C ) = E
9		decma.f	|- ( ( B x. P ) + D ) = F
10		10nn0	|- ; 1 0 e. NN0
11		dfdec10	|- ; A B = ( ( ; 1 0 x. A ) + B )
12	5 11	eqtri	|- M = ( ( ; 1 0 x. A ) + B )
13		dfdec10	|- ; C D = ( ( ; 1 0 x. C ) + D )
14	6 13	eqtri	|- N = ( ( ; 1 0 x. C ) + D )
15	10 1 2 3 4 12 14 7 8 9	numma	|- ( ( M x. P ) + N ) = ( ( ; 1 0 x. E ) + F )
16		dfdec10	|- ; E F = ( ( ; 1 0 x. E ) + F )
17	15 16	eqtr4i	|- ( ( M x. P ) + N ) = ; E F