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Theorem numma 8470
Description: Perform a multiply-add of two decimal integers  M and  N against a fixed multiplicand  P (no 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
)
numma.8  |-  P  e. 
NN0
numma.9  |-  ( ( A  x.  P )  +  C )  =  E
numma.10  |-  ( ( B  x.  P )  +  D )  =  F
Assertion
Ref Expression
numma  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)

Proof of Theorem numma
StepHypRef Expression
1 numma.6 . . . 4  |-  M  =  ( ( T  x.  A )  +  B
)
21oveq1i 5550 . . 3  |-  ( M  x.  P )  =  ( ( ( T  x.  A )  +  B )  x.  P
)
3 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
42, 3oveq12i 5552 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( ( ( T  x.  A )  +  B )  x.  P )  +  ( ( T  x.  C
)  +  D ) )
5 numma.1 . . . . . . 7  |-  T  e. 
NN0
65nn0cni 8251 . . . . . 6  |-  T  e.  CC
7 numma.2 . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 8251 . . . . . . 7  |-  A  e.  CC
9 numma.8 . . . . . . . 8  |-  P  e. 
NN0
109nn0cni 8251 . . . . . . 7  |-  P  e.  CC
118, 10mulcli 7090 . . . . . 6  |-  ( A  x.  P )  e.  CC
12 numma.4 . . . . . . 7  |-  C  e. 
NN0
1312nn0cni 8251 . . . . . 6  |-  C  e.  CC
146, 11, 13adddii 7095 . . . . 5  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
156, 8, 10mulassi 7094 . . . . . 6  |-  ( ( T  x.  A )  x.  P )  =  ( T  x.  ( A  x.  P )
)
1615oveq1i 5550 . . . . 5  |-  ( ( ( T  x.  A
)  x.  P )  +  ( T  x.  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
1714, 16eqtr4i 2079 . . . 4  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( ( T  x.  A )  x.  P )  +  ( T  x.  C ) )
1817oveq1i 5550 . . 3  |-  ( ( T  x.  ( ( A  x.  P )  +  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( ( ( T  x.  A )  x.  P )  +  ( T  x.  C
) )  +  ( ( B  x.  P
)  +  D ) )
196, 8mulcli 7090 . . . . . 6  |-  ( T  x.  A )  e.  CC
20 numma.3 . . . . . . 7  |-  B  e. 
NN0
2120nn0cni 8251 . . . . . 6  |-  B  e.  CC
2219, 21, 10adddiri 7096 . . . . 5  |-  ( ( ( T  x.  A
)  +  B )  x.  P )  =  ( ( ( T  x.  A )  x.  P )  +  ( B  x.  P ) )
2322oveq1i 5550 . . . 4  |-  ( ( ( ( T  x.  A )  +  B
)  x.  P )  +  ( ( T  x.  C )  +  D ) )  =  ( ( ( ( T  x.  A )  x.  P )  +  ( B  x.  P
) )  +  ( ( T  x.  C
)  +  D ) )
2419, 10mulcli 7090 . . . . 5  |-  ( ( T  x.  A )  x.  P )  e.  CC
256, 13mulcli 7090 . . . . 5  |-  ( T  x.  C )  e.  CC
2621, 10mulcli 7090 . . . . 5  |-  ( B  x.  P )  e.  CC
27 numma.5 . . . . . 6  |-  D  e. 
NN0
2827nn0cni 8251 . . . . 5  |-  D  e.  CC
2924, 25, 26, 28add4i 7239 . . . 4  |-  ( ( ( ( T  x.  A )  x.  P
)  +  ( T  x.  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( ( ( T  x.  A )  x.  P )  +  ( B  x.  P
) )  +  ( ( T  x.  C
)  +  D ) )
3023, 29eqtr4i 2079 . . 3  |-  ( ( ( ( T  x.  A )  +  B
)  x.  P )  +  ( ( T  x.  C )  +  D ) )  =  ( ( ( ( T  x.  A )  x.  P )  +  ( T  x.  C
) )  +  ( ( B  x.  P
)  +  D ) )
3118, 30eqtr4i 2079 . 2  |-  ( ( T  x.  ( ( A  x.  P )  +  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( ( ( T  x.  A )  +  B )  x.  P )  +  ( ( T  x.  C
)  +  D ) )
32 numma.9 . . . 4  |-  ( ( A  x.  P )  +  C )  =  E
3332oveq2i 5551 . . 3  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( T  x.  E
)
34 numma.10 . . 3  |-  ( ( B  x.  P )  +  D )  =  F
3533, 34oveq12i 5552 . 2  |-  ( ( T  x.  ( ( A  x.  P )  +  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( T  x.  E )  +  F
)
364, 31, 353eqtr2i 2082 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409  (class class class)co 5540    + caddc 6950    x. cmul 6952   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-cnex 7033  ax-resscn 7034  ax-1re 7036  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-rnegex 7051
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-iota 4895  df-fv 4938  df-ov 5543  df-inn 7991  df-n0 8240
This theorem is referenced by:  nummac  8471  numadd  8473  decma  8477
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