Theorem nummac 9387

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 9147	. . . 4 |- T e. CC
3		numma.2	. . . . . . . . 9 |- A e. NN0
4	3	nn0cni 9147	. . . . . . . 8 |- A e. CC
5		nummac.8	. . . . . . . . 9 |- P e. NN0
6	5	nn0cni 9147	. . . . . . . 8 |- P e. CC
7	4, 6	mulcli 7925	. . . . . . 7 |- ( A x. P ) e. CC
8		numma.4	. . . . . . . 8 |- C e. NN0
9	8	nn0cni 9147	. . . . . . 7 |- C e. CC
10		nummac.10	. . . . . . . 8 |- G e. NN0
11	10	nn0cni 9147	. . . . . . 7 |- G e. CC
12	7, 9, 11	addassi 7928	. . . . . 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 2191	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) = E
15	7, 9	addcli 7924	. . . . . 6 |- ( ( A x. P ) + C ) e. CC
16	15, 11	addcli 7924	. . . . 5 |- ( ( ( A x. P ) + C ) + G ) e. CC
17	14, 16	eqeltrri 2244	. . . 4 |- E e. CC
18	2, 17, 11	subdii 8326	. . 3 |- ( T x. ( E - G ) ) = ( ( T x. E ) - ( T x. G ) )
19	18	oveq1i 5863	. 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 8207	. . . . 5 |- ( ( E - G ) = ( ( A x. P ) + C ) <-> ( ( ( A x. P ) + C ) + G ) = E )
25	14, 24	mpbir 145	. . . 4 |- ( E - G ) = ( ( A x. P ) + C )
26	25	eqcomi 2174	. . 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 9386	. 2 |- ( ( M x. P ) + N ) = ( ( T x. ( E - G ) ) + ( ( T x. G ) + F ) )
29	2, 17	mulcli 7925	. . . . 5 |- ( T x. E ) e. CC
30	2, 11	mulcli 7925	. . . . 5 |- ( T x. G ) e. CC
31		npcan 8128	. . . . 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 424	. . . 4 |- ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) = ( T x. E )
33	32	oveq1i 5863	. . 3 |- ( ( ( ( T x. E ) - ( T x. G ) ) + ( T x. G ) ) + F ) = ( ( T x. E ) + F )
34	29, 30	subcli 8195	. . . 4 |- ( ( T x. E ) - ( T x. G ) ) e. CC
35		nummac.9	. . . . 5 |- F e. NN0
36	35	nn0cni 9147	. . . 4 |- F e. CC
37	34, 30, 36	addassi 7928	. . 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 2193	. 2 |- ( ( T x. E ) + F ) = ( ( ( T x. E ) - ( T x. G ) ) + ( ( T x. G ) + F ) )
39	19, 28, 38	3eqtr4i 2201	1 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779   - cmin 8090   NN0cn0 9135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-inn 8879  df-n0 9136

This theorem is referenced by:  numma2c  9388  numaddc  9390  nummul1c  9391  decmac  9394