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Theorem decmul1 9477
Description: The product of a numeral with a number (no carry). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
Hypotheses
Ref Expression
decmul1.p  |-  P  e. 
NN0
decmul1.a  |-  A  e. 
NN0
decmul1.b  |-  B  e. 
NN0
decmul1.n  |-  N  = ; A B
decmul1.0  |-  D  e. 
NN0
decmul1.c  |-  ( A  x.  P )  =  C
decmul1.d  |-  ( B  x.  P )  =  D
Assertion
Ref Expression
decmul1  |-  ( N  x.  P )  = ; C D

Proof of Theorem decmul1
StepHypRef Expression
1 10nn0 9431 . . 3  |- ; 1 0  e.  NN0
2 decmul1.p . . 3  |-  P  e. 
NN0
3 decmul1.a . . 3  |-  A  e. 
NN0
4 decmul1.b . . 3  |-  B  e. 
NN0
5 decmul1.n . . . 4  |-  N  = ; A B
6 dfdec10 9417 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
75, 6eqtri 2210 . . 3  |-  N  =  ( (; 1 0  x.  A
)  +  B )
8 decmul1.0 . . 3  |-  D  e. 
NN0
9 0nn0 9221 . . 3  |-  0  e.  NN0
103, 2nn0mulcli 9244 . . . . . 6  |-  ( A  x.  P )  e. 
NN0
1110nn0cni 9218 . . . . 5  |-  ( A  x.  P )  e.  CC
1211addid1i 8129 . . . 4  |-  ( ( A  x.  P )  +  0 )  =  ( A  x.  P
)
13 decmul1.c . . . 4  |-  ( A  x.  P )  =  C
1412, 13eqtri 2210 . . 3  |-  ( ( A  x.  P )  +  0 )  =  C
15 decmul1.d . . . . 5  |-  ( B  x.  P )  =  D
1615oveq2i 5907 . . . 4  |-  ( 0  +  ( B  x.  P ) )  =  ( 0  +  D
)
174, 2nn0mulcli 9244 . . . . . 6  |-  ( B  x.  P )  e. 
NN0
1817nn0cni 9218 . . . . 5  |-  ( B  x.  P )  e.  CC
1918addid2i 8130 . . . 4  |-  ( 0  +  ( B  x.  P ) )  =  ( B  x.  P
)
201nn0cni 9218 . . . . . . 7  |- ; 1 0  e.  CC
2120mul01i 8378 . . . . . 6  |-  (; 1 0  x.  0 )  =  0
2221eqcomi 2193 . . . . 5  |-  0  =  (; 1 0  x.  0 )
2322oveq1i 5906 . . . 4  |-  ( 0  +  D )  =  ( (; 1 0  x.  0 )  +  D )
2416, 19, 233eqtr3i 2218 . . 3  |-  ( B  x.  P )  =  ( (; 1 0  x.  0 )  +  D )
251, 2, 3, 4, 7, 8, 9, 14, 24nummul1c 9462 . 2  |-  ( N  x.  P )  =  ( (; 1 0  x.  C
)  +  D )
26 dfdec10 9417 . 2  |- ; C D  =  ( (; 1 0  x.  C
)  +  D )
2725, 26eqtr4i 2213 1  |-  ( N  x.  P )  = ; C D
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160  (class class class)co 5896   0cc0 7841   1c1 7842    + caddc 7844    x. cmul 7846   NN0cn0 9206  ;cdc 9414
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 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952
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 5899  df-oprab 5900  df-mpo 5901  df-sub 8160  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-5 9011  df-6 9012  df-7 9013  df-8 9014  df-9 9015  df-n0 9207  df-dec 9415
This theorem is referenced by:  sq10  10724
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