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Theorem nummul2c 9254
Description: The product of a decimal integer with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.)
Hypotheses
Ref Expression
nummul1c.1  |-  T  e. 
NN0
nummul1c.2  |-  P  e. 
NN0
nummul1c.3  |-  A  e. 
NN0
nummul1c.4  |-  B  e. 
NN0
nummul1c.5  |-  N  =  ( ( T  x.  A )  +  B
)
nummul1c.6  |-  D  e. 
NN0
nummul1c.7  |-  E  e. 
NN0
nummul2c.7  |-  ( ( P  x.  A )  +  E )  =  C
nummul2c.8  |-  ( P  x.  B )  =  ( ( T  x.  E )  +  D
)
Assertion
Ref Expression
nummul2c  |-  ( P  x.  N )  =  ( ( T  x.  C )  +  D
)

Proof of Theorem nummul2c
StepHypRef Expression
1 nummul1c.5 . . . 4  |-  N  =  ( ( T  x.  A )  +  B
)
2 nummul1c.1 . . . . 5  |-  T  e. 
NN0
3 nummul1c.3 . . . . 5  |-  A  e. 
NN0
4 nummul1c.4 . . . . 5  |-  B  e. 
NN0
52, 3, 4numcl 9217 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2213 . . 3  |-  N  e. 
NN0
76nn0cni 9012 . 2  |-  N  e.  CC
8 nummul1c.2 . . 3  |-  P  e. 
NN0
98nn0cni 9012 . 2  |-  P  e.  CC
10 nummul1c.6 . . 3  |-  D  e. 
NN0
11 nummul1c.7 . . 3  |-  E  e. 
NN0
123nn0cni 9012 . . . . . 6  |-  A  e.  CC
1312, 9mulcomi 7795 . . . . 5  |-  ( A  x.  P )  =  ( P  x.  A
)
1413oveq1i 5791 . . . 4  |-  ( ( A  x.  P )  +  E )  =  ( ( P  x.  A )  +  E
)
15 nummul2c.7 . . . 4  |-  ( ( P  x.  A )  +  E )  =  C
1614, 15eqtri 2161 . . 3  |-  ( ( A  x.  P )  +  E )  =  C
174nn0cni 9012 . . . 4  |-  B  e.  CC
18 nummul2c.8 . . . 4  |-  ( P  x.  B )  =  ( ( T  x.  E )  +  D
)
199, 17, 18mulcomli 7796 . . 3  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
202, 8, 3, 4, 1, 10, 11, 16, 19nummul1c 9253 . 2  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
217, 9, 20mulcomli 7796 1  |-  ( P  x.  N )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1332    e. wcel 1481  (class class class)co 5781    + caddc 7646    x. cmul 7648   NN0cn0 9000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-sub 7958  df-inn 8744  df-n0 9001
This theorem is referenced by:  decmul2c  9270
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