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Theorem nummul1c 9198
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 9162 . . . 4  |-  ( ( T  x.  A )  +  B )  e. 
NN0
61, 5eqeltri 2190 . . 3  |-  N  e. 
NN0
7 nummul1c.2 . . 3  |-  P  e. 
NN0
86, 7num0u 9160 . 2  |-  ( N  x.  P )  =  ( ( N  x.  P )  +  0 )
9 0nn0 8960 . . 3  |-  0  e.  NN0
102, 9num0h 9161 . . 3  |-  0  =  ( ( T  x.  0 )  +  0 )
11 nummul1c.6 . . 3  |-  D  e. 
NN0
12 nummul1c.7 . . 3  |-  E  e. 
NN0
1312nn0cni 8957 . . . . . 6  |-  E  e.  CC
1413addid2i 7873 . . . . 5  |-  ( 0  +  E )  =  E
1514oveq2i 5753 . . . 4  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  ( ( A  x.  P )  +  E
)
16 nummul1c.8 . . . 4  |-  ( ( A  x.  P )  +  E )  =  C
1715, 16eqtri 2138 . . 3  |-  ( ( A  x.  P )  +  ( 0  +  E ) )  =  C
184, 7num0u 9160 . . . 4  |-  ( B  x.  P )  =  ( ( B  x.  P )  +  0 )
19 nummul1c.9 . . . 4  |-  ( B  x.  P )  =  ( ( T  x.  E )  +  D
)
2018, 19eqtr3i 2140 . . 3  |-  ( ( B  x.  P )  +  0 )  =  ( ( T  x.  E )  +  D
)
212, 3, 4, 9, 9, 1, 10, 7, 11, 12, 17, 20nummac 9194 . 2  |-  ( ( N  x.  P )  +  0 )  =  ( ( T  x.  C )  +  D
)
228, 21eqtri 2138 1  |-  ( N  x.  P )  =  ( ( T  x.  C )  +  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465  (class class class)co 5742   0cc0 7588    + caddc 7591    x. cmul 7593   NN0cn0 8945
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-sub 7903  df-inn 8689  df-n0 8946
This theorem is referenced by:  nummul2c  9199  decmul1  9213  decmul1c  9214
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