Theorem nummac 9492

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 9252	. . . 4 |- T e. CC
3		numma.2	. . . . . . . . 9 |- A e. NN0
4	3	nn0cni 9252	. . . . . . . 8 |- A e. CC
5		nummac.8	. . . . . . . . 9 |- P e. NN0
6	5	nn0cni 9252	. . . . . . . 8 |- P e. CC
7	4, 6	mulcli 8024	. . . . . . 7 |- ( A x. P ) e. CC
8		numma.4	. . . . . . . 8 |- C e. NN0
9	8	nn0cni 9252	. . . . . . 7 |- C e. CC
10		nummac.10	. . . . . . . 8 |- G e. NN0
11	10	nn0cni 9252	. . . . . . 7 |- G e. CC
12	7, 9, 11	addassi 8027	. . . . . 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 2214	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) = E
15	7, 9	addcli 8023	. . . . . 6 |- ( ( A x. P ) + C ) e. CC
16	15, 11	addcli 8023	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) e. CC
17	14, 16	eqeltrri 2267	. . . 4 |- E e. CC
18	2, 17, 11	subdii 8426	. . 3 |- ( T x. ( E - G ) ) = ( ( T x. E ) - ( T x. G ) )
19	18	oveq1i 5928	. 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 8307	. . . . 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 2197	. . 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 9491	. 2 |- ( ( M x. P ) + N ) = ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) )
29	2, 17	mulcli 8024	. . . . 5 |- ( T x. E ) e. CC
30	2, 11	mulcli 8024	. . . . 5 |- ( T x. G ) e. CC
31		npcan 8228	. . . . 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 5928	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( T x. E ) + F )
34	29, 30	subcli 8295	. . . 4 |- ( ( T x. E ) - ( T x. G ) ) e. CC
35		nummac.9	. . . . 5 |- F e. NN0
36	35	nn0cni 9252	. . . 4 |- F e. CC
37	34, 30, 36	addassi 8027	. . 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 2216	. 2 |- ( ( T x. E ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
39	19, 28, 38	3eqtr4i 2224	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1364   e. wcel 2164  (class class class)co 5918   CCcc 7870   + caddc 7875   x. cmul 7877   - cmin 8190   NN0cn0 9240

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-sub 8192  df-inn 8983  df-n0 9241

This theorem is referenced by:  numma2c  9493  numaddc  9495  nummul1c  9496  decmac  9499