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Metamath Proof Explorer

Theorem numma2c

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)

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

Proof

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