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Theorem nummul1c 9242
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 9206 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2212 . . 3  |-  N  e. 
NN0
7 nummul1c.2 . . 3  |-  P  e. 
NN0
86, 7num0u 9204 . 2  |-  ( N  x.  P )  =  ( ( N  x.  P )  +  0 )
9 0nn0 9004 . . 3  |-  0  e.  NN0
102, 9num0h 9205 . . 3  |-  0  =  ( ( T  x.  0 )  +  0 )
11 nummul1c.6 . . 3  |-  D  e. 
NN0
12 nummul1c.7 . . 3  |-  E  e. 
NN0
1312nn0cni 9001 . . . . . 6  |-  E  e.  CC
1413addid2i 7917 . . . . 5  |-  ( 0  +  E )  =  E
1514oveq2i 5785 . . . 4  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  ( ( A  x.  P )  +  E
)
16 nummul1c.8 . . . 4  |-  ( ( A  x.  P )  +  E )  =  C
1715, 16eqtri 2160 . . 3  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  C
184, 7num0u 9204 . . . 4  |-  ( B  x.  P )  =  ( ( B  x.  P )  +  0 )
19 nummul1c.9 . . . 4  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
2018, 19eqtr3i 2162 . . 3  |-  ( ( B  x.  P )  +  0 )  =  ( ( T  x.  E )  +  D
)
212, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20nummac 9238 . 2  |-  ( ( N  x.  P )  +  0 )  =  ( ( T  x.  C )  +  D
)
228, 21eqtri 2160 1  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774   0cc0 7632    + caddc 7635    x. cmul 7637   NN0cn0 8989
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7947  df-inn 8733  df-n0 8990
This theorem is referenced by:  nummul2c  9243  decmul1  9257  decmul1c  9258
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