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Theorem numma2c 10416
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 10234 . . . 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 10394 . . . . . 6  |-  ( ( T  x.  A )  +  B )  e. 
NN0
83, 7eqeltri 2507 . . . . 5  |-  M  e. 
NN0
98nn0cni 10234 . . . 4  |-  M  e.  CC
102, 9mulcomi 9097 . . 3  |-  ( P  x.  M )  =  ( M  x.  P
)
1110oveq1i 6092 . 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 10234 . . . . . 6  |-  A  e.  CC
1817, 2mulcomi 9097 . . . . 5  |-  ( A  x.  P )  =  ( P  x.  A
)
1918oveq1i 6092 . . . 4  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  ( ( P  x.  A )  +  ( C  +  G ) )
20 numma2c.11 . . . 4  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
2119, 20eqtri 2457 . . 3  |-  ( ( A  x.  P )  +  ( C  +  G ) )  =  E
226nn0cni 10234 . . . . . 6  |-  B  e.  CC
2322, 2mulcomi 9097 . . . . 5  |-  ( B  x.  P )  =  ( P  x.  B
)
2423oveq1i 6092 . . . 4  |-  ( ( B  x.  P )  +  D )  =  ( ( P  x.  B )  +  D
)
25 numma2c.12 . . . 4  |-  ( ( P  x.  B )  +  D )  =  ( ( T  x.  G )  +  F
)
2624, 25eqtri 2457 . . 3  |-  ( ( B  x.  P )  +  D )  =  ( ( T  x.  G )  +  F
)
274, 5, 6, 12, 13, 3, 14, 1, 15, 16, 21, 26nummac 10415 . 2  |-  ( ( M  x.  P )  +  N )  =  ( ( T  x.  E )  +  F
)
2811, 27eqtri 2457 1  |-  ( ( P  x.  M )  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726  (class class class)co 6082    + caddc 8994    x. cmul 8996   NN0cn0 10222
This theorem is referenced by:  decma2c  10423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-ltxr 9126  df-sub 9294  df-nn 10002  df-n0 10223
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