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Theorem numma 9237
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 5784 . . 3  |-  ( M  x.  P )  =  ( ( ( T  x.  A )  +  B )  x.  P
)
3 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
42, 3oveq12i 5786 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( ( ( T  x.  A )  +  B )  x.  P )  +  ( ( T  x.  C
)  +  D ) )
5 numma.1 . . . . . . 7  |-  T  e. 
NN0
65nn0cni 9001 . . . . . 6  |-  T  e.  CC
7 numma.2 . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 9001 . . . . . . 7  |-  A  e.  CC
9 numma.8 . . . . . . . 8  |-  P  e. 
NN0
109nn0cni 9001 . . . . . . 7  |-  P  e.  CC
118, 10mulcli 7783 . . . . . 6  |-  ( A  x.  P )  e.  CC
12 numma.4 . . . . . . 7  |-  C  e. 
NN0
1312nn0cni 9001 . . . . . 6  |-  C  e.  CC
146, 11, 13adddii 7788 . . . . 5  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
156, 8, 10mulassi 7787 . . . . . 6  |-  ( ( T  x.  A )  x.  P )  =  ( T  x.  ( A  x.  P )
)
1615oveq1i 5784 . . . . 5  |-  ( ( ( T  x.  A
)  x.  P )  +  ( T  x.  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
1714, 16eqtr4i 2163 . . . 4  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( ( T  x.  A )  x.  P )  +  ( T  x.  C ) )
1817oveq1i 5784 . . 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 7783 . . . . . 6  |-  ( T  x.  A )  e.  CC
20 numma.3 . . . . . . 7  |-  B  e. 
NN0
2120nn0cni 9001 . . . . . 6  |-  B  e.  CC
2219, 21, 10adddiri 7789 . . . . 5  |-  ( ( ( T  x.  A
)  +  B )  x.  P )  =  ( ( ( T  x.  A )  x.  P )  +  ( B  x.  P ) )
2322oveq1i 5784 . . . 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 7783 . . . . 5  |-  ( ( T  x.  A )  x.  P )  e.  CC
256, 13mulcli 7783 . . . . 5  |-  ( T  x.  C )  e.  CC
2621, 10mulcli 7783 . . . . 5  |-  ( B  x.  P )  e.  CC
27 numma.5 . . . . . 6  |-  D  e. 
NN0
2827nn0cni 9001 . . . . 5  |-  D  e.  CC
2924, 25, 26, 28add4i 7939 . . . 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 2163 . . 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 2163 . 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 5785 . . 3  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( T  x.  E
)
34 numma.10 . . 3  |-  ( ( B  x.  P )  +  D )  =  F
3533, 34oveq12i 5786 . 2  |-  ( ( T  x.  ( ( A  x.  P )  +  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( T  x.  E )  +  F
)
364, 31, 353eqtr2i 2166 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774    + caddc 7635    x. cmul 7637   NN0cn0 8989
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-cnex 7723  ax-resscn 7724  ax-1re 7726  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-rnegex 7741
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-inn 8733  df-n0 8990
This theorem is referenced by:  nummac  9238  numadd  9240  decma  9244
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