Theorem nummac 9501

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 9261	. . . 4 |- T e. CC
3		numma.2	. . . . . . . . 9 |- A e. NN0
4	3	nn0cni 9261	. . . . . . . 8 |- A e. CC
5		nummac.8	. . . . . . . . 9 |- P e. NN0
6	5	nn0cni 9261	. . . . . . . 8 |- P e. CC
7	4, 6	mulcli 8031	. . . . . . 7 |- ( A x. P ) e. CC
8		numma.4	. . . . . . . 8 |- C e. NN0
9	8	nn0cni 9261	. . . . . . 7 |- C e. CC
10		nummac.10	. . . . . . . 8 |- G e. NN0
11	10	nn0cni 9261	. . . . . . 7 |- G e. CC
12	7, 9, 11	addassi 8034	. . . . . 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 2217	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) = E
15	7, 9	addcli 8030	. . . . . 6 |- ( ( A x. P ) + C ) e. CC
16	15, 11	addcli 8030	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) e. CC
17	14, 16	eqeltrri 2270	. . . 4 |- E e. CC
18	2, 17, 11	subdii 8433	. . 3 |- ( T x. ( E - G ) ) = ( ( T x. E ) - ( T x. G ) )
19	18	oveq1i 5932	. 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 8314	. . . . 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 2200	. . 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 9500	. 2 |- ( ( M x. P ) + N ) = ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) )
29	2, 17	mulcli 8031	. . . . 5 |- ( T x. E ) e. CC
30	2, 11	mulcli 8031	. . . . 5 |- ( T x. G ) e. CC
31		npcan 8235	. . . . 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 5932	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( T x. E ) + F )
34	29, 30	subcli 8302	. . . 4 |- ( ( T x. E ) - ( T x. G ) ) e. CC
35		nummac.9	. . . . 5 |- F e. NN0
36	35	nn0cni 9261	. . . 4 |- F e. CC
37	34, 30, 36	addassi 8034	. . 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 2219	. 2 |- ( ( T x. E ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
39	19, 28, 38	3eqtr4i 2227	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   + caddc 7882   x. cmul 7884   - cmin 8197   NN0cn0 9249

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-sub 8199  df-inn 8991  df-n0 9250

This theorem is referenced by:  numma2c  9502  numaddc  9504  nummul1c  9505  decmac  9508