MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nummac Structured version   Unicode version

Theorem nummac 11051
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 10848 . . . 4  |-  T  e.  CC
3 numma.2 . . . . . . . . 9  |-  A  e. 
NN0
43nn0cni 10848 . . . . . . . 8  |-  A  e.  CC
5 nummac.8 . . . . . . . . 9  |-  P  e. 
NN0
65nn0cni 10848 . . . . . . . 8  |-  P  e.  CC
74, 6mulcli 9631 . . . . . . 7  |-  ( A  x.  P )  e.  CC
8 numma.4 . . . . . . . 8  |-  C  e. 
NN0
98nn0cni 10848 . . . . . . 7  |-  C  e.  CC
10 nummac.10 . . . . . . . 8  |-  G  e. 
NN0
1110nn0cni 10848 . . . . . . 7  |-  G  e.  CC
127, 9, 11addassi 9634 . . . . . 6  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  ( ( A  x.  P )  +  ( C  +  G ) )
13 nummac.11 . . . . . 6  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
1412, 13eqtri 2431 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  =  E
157, 9addcli 9630 . . . . . 6  |-  ( ( A  x.  P )  +  C )  e.  CC
1615, 11addcli 9630 . . . . 5  |-  ( ( ( A  x.  P
)  +  C )  +  G )  e.  CC
1714, 16eqeltrri 2487 . . . 4  |-  E  e.  CC
182, 17, 11subdii 10046 . . 3  |-  ( T  x.  ( E  -  G ) )  =  ( ( T  x.  E )  -  ( T  x.  G )
)
1918oveq1i 6288 . 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 9944 . . . . 5  |-  ( ( E  -  G )  =  ( ( A  x.  P )  +  C )  <->  ( (
( A  x.  P
)  +  C )  +  G )  =  E )
2514, 24mpbir 209 . . . 4  |-  ( E  -  G )  =  ( ( A  x.  P )  +  C
)
2625eqcomi 2415 . . 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 11050 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  ( E  -  G
) )  +  ( ( T  x.  G
)  +  F ) )
292, 17mulcli 9631 . . . . 5  |-  ( T  x.  E )  e.  CC
302, 11mulcli 9631 . . . . 5  |-  ( T  x.  G )  e.  CC
31 npcan 9865 . . . . 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 670 . . . 4  |-  ( ( ( T  x.  E
)  -  ( T  x.  G ) )  +  ( T  x.  G ) )  =  ( T  x.  E
)
3332oveq1i 6288 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( T  x.  E )  +  F
)
3429, 30subcli 9931 . . . 4  |-  ( ( T  x.  E )  -  ( T  x.  G ) )  e.  CC
35 nummac.9 . . . . 5  |-  F  e. 
NN0
3635nn0cni 10848 . . . 4  |-  F  e.  CC
3734, 30, 36addassi 9634 . . 3  |-  ( ( ( ( T  x.  E )  -  ( T  x.  G )
)  +  ( T  x.  G ) )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3833, 37eqtr3i 2433 . 2  |-  ( ( T  x.  E )  +  F )  =  ( ( ( T  x.  E )  -  ( T  x.  G
) )  +  ( ( T  x.  G
)  +  F ) )
3919, 28, 383eqtr4i 2441 1  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842  (class class class)co 6278   CCcc 9520    + caddc 9525    x. cmul 9527    - cmin 9841   NN0cn0 10836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843  df-nn 10577  df-n0 10837
This theorem is referenced by:  numma2c  11052  numaddc  11054  nummul1c  11055  decmac  11058
  Copyright terms: Public domain W3C validator