Theorem nummac 9645

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 )
nummac.8	|- P e. NN0
nummac.9	|- F e. NN0
nummac.10	|- G e. NN0
nummac.11	|- ( ( A x. P ) + ( C + G ) ) = E
nummac.12	|- ( ( B x. P ) + D ) = ( ( T x. G ) + F )

Assertion

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

Proof of Theorem nummac

Step	Hyp	Ref	Expression
1		numma.1	. . . . 5 |- T e. NN0
2	1	nn0cni 9404	. . . 4 |- T e. CC
3		numma.2	. . . . . . . . 9 |- A e. NN0
4	3	nn0cni 9404	. . . . . . . 8 |- A e. CC
5		nummac.8	. . . . . . . . 9 |- P e. NN0
6	5	nn0cni 9404	. . . . . . . 8 |- P e. CC
7	4, 6	mulcli 8174	. . . . . . 7 |- ( A x. P ) e. CC
8		numma.4	. . . . . . . 8 |- C e. NN0
9	8	nn0cni 9404	. . . . . . 7 |- C e. CC
10		nummac.10	. . . . . . . 8 |- G e. NN0
11	10	nn0cni 9404	. . . . . . 7 |- G e. CC
12	7, 9, 11	addassi 8177	. . . . . 6 |- ( ( ( A x. P ) + C ) + G ) = ( ( A x. P ) + ( C + G ) )
13		nummac.11	. . . . . 6 |- ( ( A x. P ) + ( C + G ) ) = E
14	12, 13	eqtri 2250	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) = E
15	7, 9	addcli 8173	. . . . . 6 |- ( ( A x. P ) + C ) e. CC
16	15, 11	addcli 8173	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) e. CC
17	14, 16	eqeltrri 2303	. . . 4 |- E e. CC
18	2, 17, 11	subdii 8576	. . 3 |- ( T x. ( E - G ) ) = ( ( T x. E ) - ( T x. G ) )
19	18	oveq1i 6023	. 2 |- ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
20		numma.3	. . 3 |- B e. NN0
21		numma.5	. . 3 |- D e. NN0
22		numma.6	. . 3 |- M = ( ( T x. A ) + B )
23		numma.7	. . 3 |- N = ( ( T x. C ) + D )
24	17, 11, 15	subadd2i 8457	. . . . 5 |- ( ( E - G ) = ( ( A x. P ) + C ) <-> ( ( ( A x. P ) + C ) + G ) = E )
25	14, 24	mpbir 146	. . . 4 |- ( E - G ) = ( ( A x. P ) + C )
26	25	eqcomi 2233	. . 3 |- ( ( A x. P ) + C ) = ( E - G )
27		nummac.12	. . 3 |- ( ( B x. P ) + D ) = ( ( T x. G ) + F )
28	1, 3, 20, 8, 21, 22, 23, 5, 26, 27	numma 9644	. 2 |- ( ( M x. P ) + N ) = ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) )
29	2, 17	mulcli 8174	. . . . 5 |- ( T x. E ) e. CC
30	2, 11	mulcli 8174	. . . . 5 |- ( T x. G ) e. CC
31		npcan 8378	. . . . 5 |- ( ( ( T x. E ) e. CC /\ ( T x. G ) e. CC ) -> ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) = ( T x. E ) )
32	29, 30, 31	mp2an 426	. . . 4 |- ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) = ( T x. E )
33	32	oveq1i 6023	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( T x. E ) + F )
34	29, 30	subcli 8445	. . . 4 |- ( ( T x. E ) - ( T x. G ) ) e. CC
35		nummac.9	. . . . 5 |- F e. NN0
36	35	nn0cni 9404	. . . 4 |- F e. CC
37	34, 30, 36	addassi 8177	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
38	33, 37	eqtr3i 2252	. 2 |- ( ( T x. E ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
39	19, 28, 38	3eqtr4i 2260	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   + caddc 8025   x. cmul 8027   - cmin 8340   NN0cn0 9392

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8342  df-inn 9134  df-n0 9393

This theorem is referenced by:  numma2c  9646  numaddc  9648  nummul1c  9649  decmac  9652