Theorem nummac 8399

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 8193	. . . 4 |- T e. CC
3		numma.2	. . . . . . . . 9 |- A e. NN0
4	3	nn0cni 8193	. . . . . . . 8 |- A e. CC
5		nummac.8	. . . . . . . . 9 |- P e. NN0
6	5	nn0cni 8193	. . . . . . . 8 |- P e. CC
7	4, 6	mulcli 7032	. . . . . . 7 |- ( A x. P ) e. CC
8		numma.4	. . . . . . . 8 |- C e. NN0
9	8	nn0cni 8193	. . . . . . 7 |- C e. CC
10		nummac.10	. . . . . . . 8 |- G e. NN0
11	10	nn0cni 8193	. . . . . . 7 |- G e. CC
12	7, 9, 11	addassi 7035	. . . . . 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 2060	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) = E
15	7, 9	addcli 7031	. . . . . 6 |- ( ( A x. P ) + C ) e. CC
16	15, 11	addcli 7031	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) e. CC
17	14, 16	eqeltrri 2111	. . . 4 |- E e. CC
18	2, 17, 11	subdii 7404	. . 3 |- ( T x. ( E - G ) ) = ( ( T x. E ) - ( T x. G ) )
19	18	oveq1i 5522	. 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 7299	. . . . 5 |- ( ( E - G ) = ( ( A x. P ) + C ) <-> ( ( ( A x. P ) + C ) + G ) = E )
25	14, 24	mpbir 134	. . . 4 |- ( E - G ) = ( ( A x. P ) + C )
26	25	eqcomi 2044	. . 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 8398	. 2 |- ( ( M x. P ) + N ) = ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) )
29	2, 17	mulcli 7032	. . . . 5 |- ( T x. E ) e. CC
30	2, 11	mulcli 7032	. . . . 5 |- ( T x. G ) e. CC
31		npcan 7220	. . . . 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 402	. . . 4 |- ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) = ( T x. E )
33	32	oveq1i 5522	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( T x. E ) + F )
34	29, 30	subcli 7287	. . . 4 |- ( ( T x. E ) - ( T x. G ) ) e. CC
35		nummac.9	. . . . 5 |- F e. NN0
36	35	nn0cni 8193	. . . 4 |- F e. CC
37	34, 30, 36	addassi 7035	. . 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 2062	. 2 |- ( ( T x. E ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
39	19, 28, 38	3eqtr4i 2070	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   + caddc 6892   x. cmul 6894   - cmin 7182   NN0cn0 8181

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-inn 7915  df-n0 8182

This theorem is referenced by:  numma2c  8400  numaddc  8402  nummul1c  8403  decmac  8406