Theorem numma2c 9388

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 )
numma2c.8	|- P e. NN0
numma2c.9	|- F e. NN0
numma2c.10	|- G e. NN0
numma2c.11	|- ( ( P x. A ) + ( C + G ) ) = E
numma2c.12	|- ( ( P x. B ) + D ) = ( ( T x. G ) + F )

Assertion

Ref	Expression
numma2c	|- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Proof of Theorem numma2c

Step	Hyp	Ref	Expression
1		numma2c.8	. . . . 5 |- P e. NN0
2	1	nn0cni 9147	. . . 4 |- P e. CC
3		numma.6	. . . . . 6 |- M = ( ( T x. A ) + B )
4		numma.1	. . . . . . 7 |- T e. NN0
5		numma.2	. . . . . . 7 |- A e. NN0
6		numma.3	. . . . . . 7 |- B e. NN0
7	4, 5, 6	numcl 9355	. . . . . 6 |- ( ( T x. A ) + B ) e. NN0
8	3, 7	eqeltri 2243	. . . . 5 |- M e. NN0
9	8	nn0cni 9147	. . . 4 |- M e. CC
10	2, 9	mulcomi 7926	. . 3 |- ( P x. M ) = ( M x. P )
11	10	oveq1i 5863	. 2 |- ( ( P x. M ) + N ) = ( ( M x. P ) + N )
12		numma.4	. . 3 |- C e. NN0
13		numma.5	. . 3 |- D e. NN0
14		numma.7	. . 3 |- N = ( ( T x. C ) + D )
15		numma2c.9	. . 3 |- F e. NN0
16		numma2c.10	. . 3 |- G e. NN0
17	5	nn0cni 9147	. . . . . 6 |- A e. CC
18	17, 2	mulcomi 7926	. . . . 5 |- ( A x. P ) = ( P x. A )
19	18	oveq1i 5863	. . . 4 |- ( ( A x. P ) + ( C + G ) ) = ( ( P x. A ) + ( C + G ) )
20		numma2c.11	. . . 4 |- ( ( P x. A ) + ( C + G ) ) = E
21	19, 20	eqtri 2191	. . 3 |- ( ( A x. P ) + ( C + G ) ) = E
22	6	nn0cni 9147	. . . . . 6 |- B e. CC
23	22, 2	mulcomi 7926	. . . . 5 |- ( B x. P ) = ( P x. B )
24	23	oveq1i 5863	. . . 4 |- ( ( B x. P ) + D ) = ( ( P x. B ) + D )
25		numma2c.12	. . . 4 |- ( ( P x. B ) + D ) = ( ( T x. G ) + F )
26	24, 25	eqtri 2191	. . 3 |- ( ( B x. P ) + D ) = ( ( T x. G ) + F )
27	4, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26	nummac 9387	. 2 |- ( ( M x. P ) + N ) = ( ( T x. E ) + F )
28	11, 27	eqtri 2191	1 |- ( ( P x. M ) + N ) = ( ( T x. E ) + F )

Syntax hints:   = wceq 1348   e. wcel 2141  (class class class)co 5853   + caddc 7777   x. cmul 7779   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-1rid 7881  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:  decma2c  9395