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Theorem numma2c 8400
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 8193 . . . 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 8378 . . . . . 6  |-  ( ( T  x.  A )  +  B )  e. 
NN0
83, 7eqeltri 2110 . . . . 5  |-  M  e. 
NN0
98nn0cni 8193 . . . 4  |-  M  e.  CC
102, 9mulcomi 7033 . . 3  |-  ( P  x.  M )  =  ( M  x.  P
)
1110oveq1i 5522 . 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 8193 . . . . . 6  |-  A  e.  CC
1817, 2mulcomi 7033 . . . . 5  |-  ( A  x.  P )  =  ( P  x.  A
)
1918oveq1i 5522 . . . 4  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  ( ( P  x.  A )  +  ( C  +  G ) )
20 numma2c.11 . . . 4  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
2119, 20eqtri 2060 . . 3  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
226nn0cni 8193 . . . . . 6  |-  B  e.  CC
2322, 2mulcomi 7033 . . . . 5  |-  ( B  x.  P )  =  ( P  x.  B
)
2423oveq1i 5522 . . . 4  |-  ( ( B  x.  P )  +  D )  =  ( ( P  x.  B )  +  D
)
25 numma2c.12 . . . 4  |-  ( ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F
)
2624, 25eqtri 2060 . . 3  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
274, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26nummac 8399 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
2811, 27eqtri 2060 1  |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1243    e. wcel 1393  (class class class)co 5512    + caddc 6892    x. cmul 6894   NN0cn0 8181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-mulcom 6985  ax-addass 6986  ax-mulass 6987  ax-distr 6988  ax-i2m1 6989  ax-1rid 6991  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184  df-inn 7915  df-n0 8182
This theorem is referenced by:  decma2c  8407
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