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Theorem nummac 10346
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
)
nummac.8  |-  P  e. 
NN0
nummac.9  |-  F  e. 
NN0
nummac.10  |-  G  e. 
NN0
nummac.11  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
nummac.12  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
Assertion
Ref Expression
nummac  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)

Proof of Theorem nummac
StepHypRef Expression
1 numma.1 . . . . 5  |-  T  e. 
NN0
21nn0cni 10165 . . . 4  |-  T  e.  CC
3 numma.2 . . . . . . . . 9  |-  A  e. 
NN0
43nn0cni 10165 . . . . . . . 8  |-  A  e.  CC
5 nummac.8 . . . . . . . . 9  |-  P  e. 
NN0
65nn0cni 10165 . . . . . . . 8  |-  P  e.  CC
74, 6mulcli 9028 . . . . . . 7  |-  ( A  x.  P )  e.  CC
8 numma.4 . . . . . . . 8  |-  C  e. 
NN0
98nn0cni 10165 . . . . . . 7  |-  C  e.  CC
10 nummac.10 . . . . . . . 8  |-  G  e. 
NN0
1110nn0cni 10165 . . . . . . 7  |-  G  e.  CC
127, 9, 11addassi 9031 . . . . . 6  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  ( ( A  x.  P )  +  ( C  +  G ) )
13 nummac.11 . . . . . 6  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
1412, 13eqtri 2407 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  E
157, 9addcli 9027 . . . . . 6  |-  ( ( A  x.  P )  +  C )  e.  CC
1615, 11addcli 9027 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  e.  CC
1714, 16eqeltrri 2458 . . . 4  |-  E  e.  CC
182, 17, 11subdii 9414 . . 3  |-  ( T  x.  ( E  -  G ) )  =  ( ( T  x.  E )  -  ( T  x.  G )
)
1918oveq1i 6030 . 2  |-  ( ( T  x.  ( E  -  G ) )  +  ( ( T  x.  G )  +  F ) )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
20 numma.3 . . 3  |-  B  e. 
NN0
21 numma.5 . . 3  |-  D  e. 
NN0
22 numma.6 . . 3  |-  M  =  ( ( T  x.  A )  +  B
)
23 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
2417, 11, 15subadd2i 9320 . . . . 5  |-  ( ( E  -  G )  =  ( ( A  x.  P )  +  C )  <->  ( (
( A  x.  P
)  +  C )  +  G )  =  E )
2514, 24mpbir 201 . . . 4  |-  ( E  -  G )  =  ( ( A  x.  P )  +  C
)
2625eqcomi 2391 . . 3  |-  ( ( A  x.  P )  +  C )  =  ( E  -  G
)
27 nummac.12 . . 3  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
281, 3, 20, 8, 21, 22, 23, 5, 26, 27numma 10345 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  ( E  -  G
) )  +  ( ( T  x.  G
)  +  F ) )
292, 17mulcli 9028 . . . . 5  |-  ( T  x.  E )  e.  CC
302, 11mulcli 9028 . . . . 5  |-  ( T  x.  G )  e.  CC
31 npcan 9246 . . . . 5  |-  ( ( ( T  x.  E
)  e.  CC  /\  ( T  x.  G
)  e.  CC )  ->  ( ( ( T  x.  E )  -  ( T  x.  G ) )  +  ( T  x.  G
) )  =  ( T  x.  E ) )
3229, 30, 31mp2an 654 . . . 4  |-  ( ( ( T  x.  E
)  -  ( T  x.  G ) )  +  ( T  x.  G ) )  =  ( T  x.  E
)
3332oveq1i 6030 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( T  x.  E )  +  F
)
3429, 30subcli 9308 . . . 4  |-  ( ( T  x.  E )  -  ( T  x.  G ) )  e.  CC
35 nummac.9 . . . . 5  |-  F  e. 
NN0
3635nn0cni 10165 . . . 4  |-  F  e.  CC
3734, 30, 36addassi 9031 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3833, 37eqtr3i 2409 . 2  |-  ( ( T  x.  E )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3919, 28, 383eqtr4i 2417 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717  (class class class)co 6020   CCcc 8921    + caddc 8926    x. cmul 8928    - cmin 9223   NN0cn0 10153
This theorem is referenced by:  numma2c  10347  numaddc  10349  nummul1c  10350  decmac  10353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-ltxr 9058  df-sub 9225  df-nn 9933  df-n0 10154
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