Theorem numma2c 9367

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 9126	. . . 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 9334	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
8	3, 7	eqeltri 2239	. . . . 5 |- M e. NN0
9	8	nn0cni 9126	. . . 4 |- M e. CC
10	2, 9	mulcomi 7905	. . 3 |- ( P x. M ) = ( M x. P )
11	10	oveq1i 5852	. 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 9126	. . . . . 6 |- A e. CC
18	17, 2	mulcomi 7905	. . . . 5 |- ( A x. P ) = ( P x. A )
19	18	oveq1i 5852	. . . 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 2186	. . 3 |- ( ( A x. P ) + ( C + G ) ) = E
22	6	nn0cni 9126	. . . . . 6 |- B e. CC
23	22, 2	mulcomi 7905	. . . . 5 |- ( B x. P ) = ( P x. B )
24	23	oveq1i 5852	. . . 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 2186	. . 3 |- ( ( B x. P ) + D ) = ( ( T x. G ) + F )
27	4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26	nummac 9366	. 2 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )
28	11, 27	eqtri 2186	1 |- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1343   e. wcel 2136  (class class class)co 5842   + caddc 7756   x. cmul 7758   NN0cn0 9114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-inn 8858  df-n0 9115

This theorem is referenced by:  decma2c  9374