Theorem numma2c 8522

Description: Perform a multiply-add of two decimal integers M and N against a fixed multiplicand P (with 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 )
numma2c.8	|- P e. NN0
numma2c.9	|- F e. NN0
numma2c.10	|- G e. NN0
numma2c.11	|- ( ( P x. A ) + ( C + G ) ) = E
numma2c.12	|- ( ( P x. B ) + D ) = ( ( T x. G ) + F )

Assertion

Ref	Expression
numma2c	|- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Proof of Theorem numma2c

Step	Hyp	Ref	Expression
1		numma2c.8	. . . . 5 |- P e. NN0
2	1	nn0cni 8300	. . . 4 |- P e. CC
3		numma.6	. . . . . 6 |- M = ( ( T x. A ) + B )
4		numma.1	. . . . . . 7 |- T e. NN0
5		numma.2	. . . . . . 7 |- A e. NN0
6		numma.3	. . . . . . 7 |- B e. NN0
7	4, 5, 6	numcl 8489	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
8	3, 7	eqeltri 2151	. . . . 5 |- M e. NN0
9	8	nn0cni 8300	. . . 4 |- M e. CC
10	2, 9	mulcomi 7125	. . 3 |- ( P x. M ) = ( M x. P )
11	10	oveq1i 5542	. 2 |- ( ( P x. M ) + N ) = ( ( M x. P ) + N )
12		numma.4	. . 3 |- C e. NN0
13		numma.5	. . 3 |- D e. NN0
14		numma.7	. . 3 |- N = ( ( T x. C ) + D )
15		numma2c.9	. . 3 |- F e. NN0
16		numma2c.10	. . 3 |- G e. NN0
17	5	nn0cni 8300	. . . . . 6 |- A e. CC
18	17, 2	mulcomi 7125	. . . . 5 |- ( A x. P ) = ( P x. A )
19	18	oveq1i 5542	. . . 4 |- ( ( A x. P ) + ( C + G ) ) = ( ( P x. A ) + ( C + G ) )
20		numma2c.11	. . . 4 |- ( ( P x. A ) + ( C + G ) ) = E
21	19, 20	eqtri 2101	. . 3 |- ( ( A x. P ) + ( C + G ) ) = E
22	6	nn0cni 8300	. . . . . 6 |- B e. CC
23	22, 2	mulcomi 7125	. . . . 5 |- ( B x. P ) = ( P x. B )
24	23	oveq1i 5542	. . . 4 |- ( ( B x. P ) + D ) = ( ( P x. B ) + D )
25		numma2c.12	. . . 4 |- ( ( P x. B ) + D ) = ( ( T x. G ) + F )
26	24, 25	eqtri 2101	. . 3 |- ( ( B x. P ) + D ) = ( ( T x. G ) + F )
27	4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26	nummac 8521	. 2 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )
28	11, 27	eqtri 2101	1 |- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1284   e. wcel 1433  (class class class)co 5532   + caddc 6984   x. cmul 6986   NN0cn0 8288

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-sub 7281  df-inn 8040  df-n0 8289

This theorem is referenced by:  decma2c  8529