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Theorem nummac 10787
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 10591 . . . 4  |-  T  e.  CC
3 numma.2 . . . . . . . . 9  |-  A  e. 
NN0
43nn0cni 10591 . . . . . . . 8  |-  A  e.  CC
5 nummac.8 . . . . . . . . 9  |-  P  e. 
NN0
65nn0cni 10591 . . . . . . . 8  |-  P  e.  CC
74, 6mulcli 9391 . . . . . . 7  |-  ( A  x.  P )  e.  CC
8 numma.4 . . . . . . . 8  |-  C  e. 
NN0
98nn0cni 10591 . . . . . . 7  |-  C  e.  CC
10 nummac.10 . . . . . . . 8  |-  G  e. 
NN0
1110nn0cni 10591 . . . . . . 7  |-  G  e.  CC
127, 9, 11addassi 9394 . . . . . 6  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  ( ( A  x.  P )  +  ( C  +  G ) )
13 nummac.11 . . . . . 6  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
1412, 13eqtri 2463 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  E
157, 9addcli 9390 . . . . . 6  |-  ( ( A  x.  P )  +  C )  e.  CC
1615, 11addcli 9390 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  e.  CC
1714, 16eqeltrri 2514 . . . 4  |-  E  e.  CC
182, 17, 11subdii 9793 . . 3  |-  ( T  x.  ( E  -  G ) )  =  ( ( T  x.  E )  -  ( T  x.  G )
)
1918oveq1i 6101 . 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 9696 . . . . 5  |-  ( ( E  -  G )  =  ( ( A  x.  P )  +  C )  <->  ( (
( A  x.  P
)  +  C )  +  G )  =  E )
2514, 24mpbir 209 . . . 4  |-  ( E  -  G )  =  ( ( A  x.  P )  +  C
)
2625eqcomi 2447 . . 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 10786 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  ( E  -  G
) )  +  ( ( T  x.  G
)  +  F ) )
292, 17mulcli 9391 . . . . 5  |-  ( T  x.  E )  e.  CC
302, 11mulcli 9391 . . . . 5  |-  ( T  x.  G )  e.  CC
31 npcan 9619 . . . . 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 672 . . . 4  |-  ( ( ( T  x.  E
)  -  ( T  x.  G ) )  +  ( T  x.  G ) )  =  ( T  x.  E
)
3332oveq1i 6101 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( T  x.  E )  +  F
)
3429, 30subcli 9684 . . . 4  |-  ( ( T  x.  E )  -  ( T  x.  G ) )  e.  CC
35 nummac.9 . . . . 5  |-  F  e. 
NN0
3635nn0cni 10591 . . . 4  |-  F  e.  CC
3734, 30, 36addassi 9394 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3833, 37eqtr3i 2465 . 2  |-  ( ( T  x.  E )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3919, 28, 383eqtr4i 2473 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756  (class class class)co 6091   CCcc 9280    + caddc 9285    x. cmul 9287    - cmin 9595   NN0cn0 10579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423  df-sub 9597  df-nn 10323  df-n0 10580
This theorem is referenced by:  numma2c  10788  numaddc  10790  nummul1c  10791  decmac  10794
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