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Theorem nummul1c 8475
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 8439 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2126 . . 3  |-  N  e. 
NN0
7 nummul1c.2 . . 3  |-  P  e. 
NN0
86, 7num0u 8437 . 2  |-  ( N  x.  P )  =  ( ( N  x.  P )  +  0 )
9 0nn0 8254 . . 3  |-  0  e.  NN0
102, 9num0h 8438 . . 3  |-  0  =  ( ( T  x.  0 )  +  0 )
11 nummul1c.6 . . 3  |-  D  e. 
NN0
12 nummul1c.7 . . 3  |-  E  e. 
NN0
1312nn0cni 8251 . . . . . 6  |-  E  e.  CC
1413addid2i 7217 . . . . 5  |-  ( 0  +  E )  =  E
1514oveq2i 5551 . . . 4  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  ( ( A  x.  P )  +  E
)
16 nummul1c.8 . . . 4  |-  ( ( A  x.  P )  +  E )  =  C
1715, 16eqtri 2076 . . 3  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  C
184, 7num0u 8437 . . . 4  |-  ( B  x.  P )  =  ( ( B  x.  P )  +  0 )
19 nummul1c.9 . . . 4  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
2018, 19eqtr3i 2078 . . 3  |-  ( ( B  x.  P )  +  0 )  =  ( ( T  x.  E )  +  D
)
212, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20nummac 8471 . 2  |-  ( ( N  x.  P )  +  0 )  =  ( ( T  x.  C )  +  D
)
228, 21eqtri 2076 1  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1259    e. wcel 1409  (class class class)co 5540   0cc0 6947    + caddc 6950    x. cmul 6952   NN0cn0 8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-cnre 7053
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-sub 7247  df-inn 7991  df-n0 8240
This theorem is referenced by:  nummul2c  8476  decmul1  8490  decmul1c  8491
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