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Theorem nummac 9387
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 9147 . . . 4  |-  T  e.  CC
3 numma.2 . . . . . . . . 9  |-  A  e. 
NN0
43nn0cni 9147 . . . . . . . 8  |-  A  e.  CC
5 nummac.8 . . . . . . . . 9  |-  P  e. 
NN0
65nn0cni 9147 . . . . . . . 8  |-  P  e.  CC
74, 6mulcli 7925 . . . . . . 7  |-  ( A  x.  P )  e.  CC
8 numma.4 . . . . . . . 8  |-  C  e. 
NN0
98nn0cni 9147 . . . . . . 7  |-  C  e.  CC
10 nummac.10 . . . . . . . 8  |-  G  e. 
NN0
1110nn0cni 9147 . . . . . . 7  |-  G  e.  CC
127, 9, 11addassi 7928 . . . . . 6  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  ( ( A  x.  P )  +  ( C  +  G ) )
13 nummac.11 . . . . . 6  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
1412, 13eqtri 2191 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  E
157, 9addcli 7924 . . . . . 6  |-  ( ( A  x.  P )  +  C )  e.  CC
1615, 11addcli 7924 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  e.  CC
1714, 16eqeltrri 2244 . . . 4  |-  E  e.  CC
182, 17, 11subdii 8326 . . 3  |-  ( T  x.  ( E  -  G ) )  =  ( ( T  x.  E )  -  ( T  x.  G )
)
1918oveq1i 5863 . 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 8207 . . . . 5  |-  ( ( E  -  G )  =  ( ( A  x.  P )  +  C )  <->  ( (
( A  x.  P
)  +  C )  +  G )  =  E )
2514, 24mpbir 145 . . . 4  |-  ( E  -  G )  =  ( ( A  x.  P )  +  C
)
2625eqcomi 2174 . . 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 9386 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  ( E  -  G
) )  +  ( ( T  x.  G
)  +  F ) )
292, 17mulcli 7925 . . . . 5  |-  ( T  x.  E )  e.  CC
302, 11mulcli 7925 . . . . 5  |-  ( T  x.  G )  e.  CC
31 npcan 8128 . . . . 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 424 . . . 4  |-  ( ( ( T  x.  E
)  -  ( T  x.  G ) )  +  ( T  x.  G ) )  =  ( T  x.  E
)
3332oveq1i 5863 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( T  x.  E )  +  F
)
3429, 30subcli 8195 . . . 4  |-  ( ( T  x.  E )  -  ( T  x.  G ) )  e.  CC
35 nummac.9 . . . . 5  |-  F  e. 
NN0
3635nn0cni 9147 . . . 4  |-  F  e.  CC
3734, 30, 36addassi 7928 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3833, 37eqtr3i 2193 . 2  |-  ( ( T  x.  E )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3919, 28, 383eqtr4i 2201 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141  (class class class)co 5853   CCcc 7772    + caddc 7777    x. cmul 7779    - cmin 8090   NN0cn0 9135
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-inn 8879  df-n0 9136
This theorem is referenced by:  numma2c  9388  numaddc  9390  nummul1c  9391  decmac  9394
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