Theorem numma 9019

Description: Perform a multiply-add of two decimal integers M and N against a fixed multiplicand P (no carry). (Contributed by Mario Carneiro, 18-Feb-2014.)

Hypotheses

Ref	Expression
numma.1	|- T e. NN0
numma.2	|- A e. NN0
numma.3	|- B e. NN0
numma.4	|- C e. NN0
numma.5	|- D e. NN0
numma.6	|- M = ( ( T x. A ) + B )
numma.7	|- N = ( ( T x. C ) + D )
numma.8	|- P e. NN0
numma.9	|- ( ( A x. P ) + C ) = E
numma.10	|- ( ( B x. P ) + D ) = F

Assertion

Ref	Expression
numma	|- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Proof of Theorem numma

Step	Hyp	Ref	Expression
1		numma.6	. . . 4 |- M = ( ( T x. A ) + B )
2	1	oveq1i 5700	. . 3 |- ( M x. P ) = ( ( ( T x. A ) + B ) x. P )
3		numma.7	. . 3 |- N = ( ( T x. C ) + D )
4	2, 3	oveq12i 5702	. 2 |- ( ( M x. P ) + N ) = ( ( ( ( T x. A ) + B ) x. P ) + ( ( T x. C ) + D ) )
5		numma.1	. . . . . . 7 |- T e. NN0
6	5	nn0cni 8783	. . . . . 6 |- T e. CC
7		numma.2	. . . . . . . 8 |- A e. NN0
8	7	nn0cni 8783	. . . . . . 7 |- A e. CC
9		numma.8	. . . . . . . 8 |- P e. NN0
10	9	nn0cni 8783	. . . . . . 7 |- P e. CC
11	8, 10	mulcli 7590	. . . . . 6 |- ( A x. P ) e. CC
12		numma.4	. . . . . . 7 |- C e. NN0
13	12	nn0cni 8783	. . . . . 6 |- C e. CC
14	6, 11, 13	adddii 7595	. . . . 5 |- ( T x. ( ( A x. P ) + C ) ) = ( ( T x. ( A x. P ) ) + ( T x. C ) )
15	6, 8, 10	mulassi 7594	. . . . . 6 |- ( ( T x. A ) x. P ) = ( T x. ( A x. P ) )
16	15	oveq1i 5700	. . . . 5 |- ( ( ( T x. A ) x. P ) + ( T x. C ) ) = ( ( T x. ( A x. P ) ) + ( T x. C ) )
17	14, 16	eqtr4i 2118	. . . 4 |- ( T x. ( ( A x. P ) + C ) ) = ( ( ( T x. A ) x. P ) + ( T x. C ) )
18	17	oveq1i 5700	. . 3 |- ( ( T x. ( ( A x. P ) + C ) ) + ( ( B x. P ) + D ) ) = ( ( ( ( T x. A ) x. P ) + ( T x. C ) ) + ( ( B x. P ) + D ) )
19	6, 8	mulcli 7590	. . . . . 6 |- ( T x. A ) e. CC
20		numma.3	. . . . . . 7 |- B e. NN0
21	20	nn0cni 8783	. . . . . 6 |- B e. CC
22	19, 21, 10	adddiri 7596	. . . . 5 |- ( ( ( T x. A ) + B ) x. P ) = ( ( ( T x. A ) x. P ) + ( B x. P ) )
23	22	oveq1i 5700	. . . 4 |- ( ( ( ( T x. A ) + B ) x. P ) + ( ( T x. C ) + D ) ) = ( ( ( ( T x. A ) x. P ) + ( B x. P ) ) + ( ( T x. C ) + D ) )
24	19, 10	mulcli 7590	. . . . 5 |- ( ( T x. A ) x. P ) e. CC
25	6, 13	mulcli 7590	. . . . 5 |- ( T x. C ) e. CC
26	21, 10	mulcli 7590	. . . . 5 |- ( B x. P ) e. CC
27		numma.5	. . . . . 6 |- D e. NN0
28	27	nn0cni 8783	. . . . 5 |- D e. CC
29	24, 25, 26, 28	add4i 7744	. . . 4 |- ( ( ( ( T x. A ) x. P ) + ( T x. C ) ) + ( ( B x. P ) + D ) ) = ( ( ( ( T x. A ) x. P ) + ( B x. P ) ) + ( ( T x. C ) + D ) )
30	23, 29	eqtr4i 2118	. . 3 |- ( ( ( ( T x. A ) + B ) x. P ) + ( ( T x. C ) + D ) ) = ( ( ( ( T x. A ) x. P ) + ( T x. C ) ) + ( ( B x. P ) + D ) )
31	18, 30	eqtr4i 2118	. 2 |- ( ( T x. ( ( A x. P ) + C ) ) + ( ( B x. P ) + D ) ) = ( ( ( ( T x. A ) + B ) x. P ) + ( ( T x. C ) + D ) )
32		numma.9	. . . 4 |- ( ( A x. P ) + C ) = E
33	32	oveq2i 5701	. . 3 |- ( T x. ( ( A x. P ) + C ) ) = ( T x. E )
34		numma.10	. . 3 |- ( ( B x. P ) + D ) = F
35	33, 34	oveq12i 5702	. 2 |- ( ( T x. ( ( A x. P ) + C ) ) + ( ( B x. P ) + D ) ) = ( ( T x. E ) + F )
36	4, 31, 35	3eqtr2i 2121	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1296   e. wcel 1445  (class class class)co 5690   + caddc 7450   x. cmul 7452   NN0cn0 8771

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-cnex 7533  ax-resscn 7534  ax-1re 7536  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-rnegex 7551

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-iota 5014  df-fv 5057  df-ov 5693  df-inn 8521  df-n0 8772

This theorem is referenced by:  nummac  9020  numadd  9022  decma  9026