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Theorem numaddc 8924
Description: Add two decimal integers  M and  N (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
)
numaddc.8  |-  F  e. 
NN0
numaddc.9  |-  ( ( A  +  C )  +  1 )  =  E
numaddc.10  |-  ( B  +  D )  =  ( ( T  x.  1 )  +  F
)
Assertion
Ref Expression
numaddc  |-  ( M  +  N )  =  ( ( T  x.  E )  +  F
)

Proof of Theorem numaddc
StepHypRef Expression
1 numma.6 . . . . . 6  |-  M  =  ( ( T  x.  A )  +  B
)
2 numma.1 . . . . . . 7  |-  T  e. 
NN0
3 numma.2 . . . . . . 7  |-  A  e. 
NN0
4 numma.3 . . . . . . 7  |-  B  e. 
NN0
52, 3, 4numcl 8889 . . . . . 6  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2160 . . . . 5  |-  M  e. 
NN0
76nn0cni 8685 . . . 4  |-  M  e.  CC
87mulid1i 7490 . . 3  |-  ( M  x.  1 )  =  M
98oveq1i 5662 . 2  |-  ( ( M  x.  1 )  +  N )  =  ( M  +  N
)
10 numma.4 . . 3  |-  C  e. 
NN0
11 numma.5 . . 3  |-  D  e. 
NN0
12 numma.7 . . 3  |-  N  =  ( ( T  x.  C )  +  D
)
13 1nn0 8689 . . 3  |-  1  e.  NN0
14 numaddc.8 . . 3  |-  F  e. 
NN0
153nn0cni 8685 . . . . . 6  |-  A  e.  CC
1615mulid1i 7490 . . . . 5  |-  ( A  x.  1 )  =  A
1716oveq1i 5662 . . . 4  |-  ( ( A  x.  1 )  +  ( C  + 
1 ) )  =  ( A  +  ( C  +  1 ) )
1810nn0cni 8685 . . . . 5  |-  C  e.  CC
19 ax-1cn 7438 . . . . 5  |-  1  e.  CC
2015, 18, 19addassi 7496 . . . 4  |-  ( ( A  +  C )  +  1 )  =  ( A  +  ( C  +  1 ) )
21 numaddc.9 . . . 4  |-  ( ( A  +  C )  +  1 )  =  E
2217, 20, 213eqtr2i 2114 . . 3  |-  ( ( A  x.  1 )  +  ( C  + 
1 ) )  =  E
234nn0cni 8685 . . . . . 6  |-  B  e.  CC
2423mulid1i 7490 . . . . 5  |-  ( B  x.  1 )  =  B
2524oveq1i 5662 . . . 4  |-  ( ( B  x.  1 )  +  D )  =  ( B  +  D
)
26 numaddc.10 . . . 4  |-  ( B  +  D )  =  ( ( T  x.  1 )  +  F
)
2725, 26eqtri 2108 . . 3  |-  ( ( B  x.  1 )  +  D )  =  ( ( T  x.  1 )  +  F
)
282, 3, 4, 10, 11, 1, 12, 13, 14, 13, 22, 27nummac 8921 . 2  |-  ( ( M  x.  1 )  +  N )  =  ( ( T  x.  E )  +  F
)
299, 28eqtr3i 2110 1  |-  ( M  +  N )  =  ( ( T  x.  E )  +  F
)
Colors of variables: wff set class
Syntax hints:    = wceq 1289    e. wcel 1438  (class class class)co 5652   1c1 7351    + caddc 7353    x. cmul 7355   NN0cn0 8673
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-inn 8423  df-n0 8674
This theorem is referenced by:  decaddc  8931
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