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Theorem numma2c 9195
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
)
numma2c.8  |-  P  e. 
NN0
numma2c.9  |-  F  e. 
NN0
numma2c.10  |-  G  e. 
NN0
numma2c.11  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
numma2c.12  |-  ( ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F
)
Assertion
Ref Expression
numma2c  |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F
)

Proof of Theorem numma2c
StepHypRef Expression
1 numma2c.8 . . . . 5  |-  P  e. 
NN0
21nn0cni 8957 . . . 4  |-  P  e.  CC
3 numma.6 . . . . . 6  |-  M  =  ( ( T  x.  A )  +  B
)
4 numma.1 . . . . . . 7  |-  T  e. 
NN0
5 numma.2 . . . . . . 7  |-  A  e. 
NN0
6 numma.3 . . . . . . 7  |-  B  e. 
NN0
74, 5, 6numcl 9162 . . . . . 6  |-  ( ( T  x.  A )  +  B )  e. 
NN0
83, 7eqeltri 2190 . . . . 5  |-  M  e. 
NN0
98nn0cni 8957 . . . 4  |-  M  e.  CC
102, 9mulcomi 7740 . . 3  |-  ( P  x.  M )  =  ( M  x.  P
)
1110oveq1i 5752 . 2  |-  ( ( P  x.  M )  +  N )  =  ( ( M  x.  P )  +  N
)
12 numma.4 . . 3  |-  C  e. 
NN0
13 numma.5 . . 3  |-  D  e. 
NN0
14 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
15 numma2c.9 . . 3  |-  F  e. 
NN0
16 numma2c.10 . . 3  |-  G  e. 
NN0
175nn0cni 8957 . . . . . 6  |-  A  e.  CC
1817, 2mulcomi 7740 . . . . 5  |-  ( A  x.  P )  =  ( P  x.  A
)
1918oveq1i 5752 . . . 4  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  ( ( P  x.  A )  +  ( C  +  G ) )
20 numma2c.11 . . . 4  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
2119, 20eqtri 2138 . . 3  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
226nn0cni 8957 . . . . . 6  |-  B  e.  CC
2322, 2mulcomi 7740 . . . . 5  |-  ( B  x.  P )  =  ( P  x.  B
)
2423oveq1i 5752 . . . 4  |-  ( ( B  x.  P )  +  D )  =  ( ( P  x.  B )  +  D
)
25 numma2c.12 . . . 4  |-  ( ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F
)
2624, 25eqtri 2138 . . 3  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
274, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26nummac 9194 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
2811, 27eqtri 2138 1  |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465  (class class class)co 5742    + caddc 7591    x. cmul 7593   NN0cn0 8945
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-inn 8689  df-n0 8946
This theorem is referenced by:  decma2c  9202
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