ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nummul1c Unicode version

Theorem nummul1c 9463
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 9427 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2262 . . 3  |-  N  e. 
NN0
7 nummul1c.2 . . 3  |-  P  e. 
NN0
86, 7num0u 9425 . 2  |-  ( N  x.  P )  =  ( ( N  x.  P )  +  0 )
9 0nn0 9222 . . 3  |-  0  e.  NN0
102, 9num0h 9426 . . 3  |-  0  =  ( ( T  x.  0 )  +  0 )
11 nummul1c.6 . . 3  |-  D  e. 
NN0
12 nummul1c.7 . . 3  |-  E  e. 
NN0
1312nn0cni 9219 . . . . . 6  |-  E  e.  CC
1413addid2i 8131 . . . . 5  |-  ( 0  +  E )  =  E
1514oveq2i 5908 . . . 4  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  ( ( A  x.  P )  +  E
)
16 nummul1c.8 . . . 4  |-  ( ( A  x.  P )  +  E )  =  C
1715, 16eqtri 2210 . . 3  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  C
184, 7num0u 9425 . . . 4  |-  ( B  x.  P )  =  ( ( B  x.  P )  +  0 )
19 nummul1c.9 . . . 4  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
2018, 19eqtr3i 2212 . . 3  |-  ( ( B  x.  P )  +  0 )  =  ( ( T  x.  E )  +  D
)
212, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20nummac 9459 . 2  |-  ( ( N  x.  P )  +  0 )  =  ( ( T  x.  C )  +  D
)
228, 21eqtri 2210 1  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160  (class class class)co 5897   0cc0 7842    + caddc 7845    x. cmul 7847   NN0cn0 9207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-sub 8161  df-inn 8951  df-n0 9208
This theorem is referenced by:  nummul2c  9464  decmul1  9478  decmul1c  9479
  Copyright terms: Public domain W3C validator