Theorem numma2c 9584

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 9342	. . . 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 9551	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
8	3, 7	eqeltri 2280	. . . . 5 |- M e. NN0
9	8	nn0cni 9342	. . . 4 |- M e. CC
10	2, 9	mulcomi 8113	. . 3 |- ( P x. M ) = ( M x. P )
11	10	oveq1i 5977	. 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 9342	. . . . . 6 |- A e. CC
18	17, 2	mulcomi 8113	. . . . 5 |- ( A x. P ) = ( P x. A )
19	18	oveq1i 5977	. . . 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 2228	. . 3 |- ( ( A x. P ) + ( C + G ) ) = E
22	6	nn0cni 9342	. . . . . 6 |- B e. CC
23	22, 2	mulcomi 8113	. . . . 5 |- ( B x. P ) = ( P x. B )
24	23	oveq1i 5977	. . . 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 2228	. . 3 |- ( ( B x. P ) + D ) = ( ( T x. G ) + F )
27	4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26	nummac 9583	. 2 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )
28	11, 27	eqtri 2228	1 |- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1373   e. wcel 2178  (class class class)co 5967   + caddc 7963   x. cmul 7965   NN0cn0 9330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-inn 9072  df-n0 9331

This theorem is referenced by:  decma2c  9591