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Theorem nummac 10156
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 9977 . . . 4  |-  T  e.  CC
3 numma.2 . . . . . . . . 9  |-  A  e. 
NN0
43nn0cni 9977 . . . . . . . 8  |-  A  e.  CC
5 nummac.8 . . . . . . . . 9  |-  P  e. 
NN0
65nn0cni 9977 . . . . . . . 8  |-  P  e.  CC
74, 6mulcli 8842 . . . . . . 7  |-  ( A  x.  P )  e.  CC
8 numma.4 . . . . . . . 8  |-  C  e. 
NN0
98nn0cni 9977 . . . . . . 7  |-  C  e.  CC
10 nummac.10 . . . . . . . 8  |-  G  e. 
NN0
1110nn0cni 9977 . . . . . . 7  |-  G  e.  CC
127, 9, 11addassi 8845 . . . . . 6  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  ( ( A  x.  P )  +  ( C  +  G ) )
13 nummac.11 . . . . . 6  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
1412, 13eqtri 2303 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  E
157, 9addcli 8841 . . . . . 6  |-  ( ( A  x.  P )  +  C )  e.  CC
1615, 11addcli 8841 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  e.  CC
1714, 16eqeltrri 2354 . . . 4  |-  E  e.  CC
182, 17, 11subdii 9228 . . 3  |-  ( T  x.  ( E  -  G ) )  =  ( ( T  x.  E )  -  ( T  x.  G )
)
1918oveq1i 5868 . 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 9134 . . . . 5  |-  ( ( E  -  G )  =  ( ( A  x.  P )  +  C )  <->  ( (
( A  x.  P
)  +  C )  +  G )  =  E )
2514, 24mpbir 200 . . . 4  |-  ( E  -  G )  =  ( ( A  x.  P )  +  C
)
2625eqcomi 2287 . . 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 10155 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  ( E  -  G
) )  +  ( ( T  x.  G
)  +  F ) )
292, 17mulcli 8842 . . . . 5  |-  ( T  x.  E )  e.  CC
302, 11mulcli 8842 . . . . 5  |-  ( T  x.  G )  e.  CC
31 npcan 9060 . . . . 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 653 . . . 4  |-  ( ( ( T  x.  E
)  -  ( T  x.  G ) )  +  ( T  x.  G ) )  =  ( T  x.  E
)
3332oveq1i 5868 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( T  x.  E )  +  F
)
3429, 30subcli 9122 . . . 4  |-  ( ( T  x.  E )  -  ( T  x.  G ) )  e.  CC
35 nummac.9 . . . . 5  |-  F  e. 
NN0
3635nn0cni 9977 . . . 4  |-  F  e.  CC
3734, 30, 36addassi 8845 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3833, 37eqtr3i 2305 . 2  |-  ( ( T  x.  E )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3919, 28, 383eqtr4i 2313 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   NN0cn0 9965
This theorem is referenced by:  numma2c  10157  numaddc  10159  nummul1c  10160  decmac  10163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-nn 9747  df-n0 9966
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