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Theorem numma 10171
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 5884 . . 3  |-  ( M  x.  P )  =  ( ( ( T  x.  A )  +  B )  x.  P
)
3 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
42, 3oveq12i 5886 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( ( ( T  x.  A )  +  B )  x.  P )  +  ( ( T  x.  C
)  +  D ) )
5 numma.1 . . . . . . 7  |-  T  e. 
NN0
65nn0cni 9993 . . . . . 6  |-  T  e.  CC
7 numma.2 . . . . . . . 8  |-  A  e. 
NN0
87nn0cni 9993 . . . . . . 7  |-  A  e.  CC
9 numma.8 . . . . . . . 8  |-  P  e. 
NN0
109nn0cni 9993 . . . . . . 7  |-  P  e.  CC
118, 10mulcli 8858 . . . . . 6  |-  ( A  x.  P )  e.  CC
12 numma.4 . . . . . . 7  |-  C  e. 
NN0
1312nn0cni 9993 . . . . . 6  |-  C  e.  CC
146, 11, 13adddii 8863 . . . . 5  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
156, 8, 10mulassi 8862 . . . . . 6  |-  ( ( T  x.  A )  x.  P )  =  ( T  x.  ( A  x.  P )
)
1615oveq1i 5884 . . . . 5  |-  ( ( ( T  x.  A
)  x.  P )  +  ( T  x.  C ) )  =  ( ( T  x.  ( A  x.  P
) )  +  ( T  x.  C ) )
1714, 16eqtr4i 2319 . . . 4  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( ( ( T  x.  A )  x.  P )  +  ( T  x.  C ) )
1817oveq1i 5884 . . 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 8858 . . . . . 6  |-  ( T  x.  A )  e.  CC
20 numma.3 . . . . . . 7  |-  B  e. 
NN0
2120nn0cni 9993 . . . . . 6  |-  B  e.  CC
2219, 21, 10adddiri 8864 . . . . 5  |-  ( ( ( T  x.  A
)  +  B )  x.  P )  =  ( ( ( T  x.  A )  x.  P )  +  ( B  x.  P ) )
2322oveq1i 5884 . . . 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 8858 . . . . 5  |-  ( ( T  x.  A )  x.  P )  e.  CC
256, 13mulcli 8858 . . . . 5  |-  ( T  x.  C )  e.  CC
2621, 10mulcli 8858 . . . . 5  |-  ( B  x.  P )  e.  CC
27 numma.5 . . . . . 6  |-  D  e. 
NN0
2827nn0cni 9993 . . . . 5  |-  D  e.  CC
2924, 25, 26, 28add4i 9047 . . . 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 2319 . . 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 2319 . 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 5885 . . 3  |-  ( T  x.  ( ( A  x.  P )  +  C ) )  =  ( T  x.  E
)
34 numma.10 . . 3  |-  ( ( B  x.  P )  +  D )  =  F
3533, 34oveq12i 5886 . 2  |-  ( ( T  x.  ( ( A  x.  P )  +  C ) )  +  ( ( B  x.  P )  +  D ) )  =  ( ( T  x.  E )  +  F
)
364, 31, 353eqtr2i 2322 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696  (class class class)co 5874    + caddc 8756    x. cmul 8758   NN0cn0 9981
This theorem is referenced by:  nummac  10172  numadd  10174  decma  10178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-nn 9763  df-n0 9982
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