ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  numma2c Unicode version

Theorem numma2c 8655
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 8419 . . . 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 8622 . . . . . 6  |-  ( ( T  x.  A )  +  B )  e. 
NN0
83, 7eqeltri 2155 . . . . 5  |-  M  e. 
NN0
98nn0cni 8419 . . . 4  |-  M  e.  CC
102, 9mulcomi 7239 . . 3  |-  ( P  x.  M )  =  ( M  x.  P
)
1110oveq1i 5573 . 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 8419 . . . . . 6  |-  A  e.  CC
1817, 2mulcomi 7239 . . . . 5  |-  ( A  x.  P )  =  ( P  x.  A
)
1918oveq1i 5573 . . . 4  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  ( ( P  x.  A )  +  ( C  +  G ) )
20 numma2c.11 . . . 4  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
2119, 20eqtri 2103 . . 3  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
226nn0cni 8419 . . . . . 6  |-  B  e.  CC
2322, 2mulcomi 7239 . . . . 5  |-  ( B  x.  P )  =  ( P  x.  B
)
2423oveq1i 5573 . . . 4  |-  ( ( B  x.  P )  +  D )  =  ( ( P  x.  B )  +  D
)
25 numma2c.12 . . . 4  |-  ( ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F
)
2624, 25eqtri 2103 . . 3  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
274, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26nummac 8654 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
2811, 27eqtri 2103 1  |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434  (class class class)co 5563    + caddc 7098    x. cmul 7100   NN0cn0 8407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-sub 7400  df-inn 8159  df-n0 8408
This theorem is referenced by:  decma2c  8662
  Copyright terms: Public domain W3C validator