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Theorem nummul1c 8986
Description: The product of a decimal integer with a number. (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
nummul1c.8  |-  ( ( A  x.  P )  +  E )  =  C
nummul1c.9  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
Assertion
Ref Expression
nummul1c  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)

Proof of Theorem nummul1c
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 8950 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2161 . . 3  |-  N  e. 
NN0
7 nummul1c.2 . . 3  |-  P  e. 
NN0
86, 7num0u 8948 . 2  |-  ( N  x.  P )  =  ( ( N  x.  P )  +  0 )
9 0nn0 8749 . . 3  |-  0  e.  NN0
102, 9num0h 8949 . . 3  |-  0  =  ( ( T  x.  0 )  +  0 )
11 nummul1c.6 . . 3  |-  D  e. 
NN0
12 nummul1c.7 . . 3  |-  E  e. 
NN0
1312nn0cni 8746 . . . . . 6  |-  E  e.  CC
1413addid2i 7686 . . . . 5  |-  ( 0  +  E )  =  E
1514oveq2i 5677 . . . 4  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  ( ( A  x.  P )  +  E
)
16 nummul1c.8 . . . 4  |-  ( ( A  x.  P )  +  E )  =  C
1715, 16eqtri 2109 . . 3  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  C
184, 7num0u 8948 . . . 4  |-  ( B  x.  P )  =  ( ( B  x.  P )  +  0 )
19 nummul1c.9 . . . 4  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
2018, 19eqtr3i 2111 . . 3  |-  ( ( B  x.  P )  +  0 )  =  ( ( T  x.  E )  +  D
)
212, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20nummac 8982 . 2  |-  ( ( N  x.  P )  +  0 )  =  ( ( T  x.  C )  +  D
)
228, 21eqtri 2109 1  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1290    e. wcel 1439  (class class class)co 5666   0cc0 7411    + caddc 7414    x. cmul 7416   NN0cn0 8734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-cnre 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-sub 7716  df-inn 8484  df-n0 8735
This theorem is referenced by:  nummul2c  8987  decmul1  9001  decmul1c  9002
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