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Theorem decmul10add 9487
Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)
Hypotheses
Ref Expression
decmul10add.1  |-  A  e. 
NN0
decmul10add.2  |-  B  e. 
NN0
decmul10add.3  |-  M  e. 
NN0
decmul10add.4  |-  E  =  ( M  x.  A
)
decmul10add.5  |-  F  =  ( M  x.  B
)
Assertion
Ref Expression
decmul10add  |-  ( M  x. ; A B )  =  (; E
0  +  F )

Proof of Theorem decmul10add
StepHypRef Expression
1 dfdec10 9422 . . 3  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
21oveq2i 5911 . 2  |-  ( M  x. ; A B )  =  ( M  x.  ( (; 1
0  x.  A )  +  B ) )
3 decmul10add.3 . . . 4  |-  M  e. 
NN0
43nn0cni 9223 . . 3  |-  M  e.  CC
5 10nn0 9436 . . . . 5  |- ; 1 0  e.  NN0
6 decmul10add.1 . . . . 5  |-  A  e. 
NN0
75, 6nn0mulcli 9249 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
87nn0cni 9223 . . 3  |-  (; 1 0  x.  A
)  e.  CC
9 decmul10add.2 . . . 4  |-  B  e. 
NN0
109nn0cni 9223 . . 3  |-  B  e.  CC
114, 8, 10adddii 8002 . 2  |-  ( M  x.  ( (; 1 0  x.  A
)  +  B ) )  =  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )
125nn0cni 9223 . . . . 5  |- ; 1 0  e.  CC
136nn0cni 9223 . . . . 5  |-  A  e.  CC
144, 12, 13mul12i 8138 . . . 4  |-  ( M  x.  (; 1 0  x.  A
) )  =  (; 1
0  x.  ( M  x.  A ) )
153, 6nn0mulcli 9249 . . . . 5  |-  ( M  x.  A )  e. 
NN0
1615dec0u 9439 . . . 4  |-  (; 1 0  x.  ( M  x.  A )
)  = ; ( M  x.  A
) 0
17 decmul10add.4 . . . . . 6  |-  E  =  ( M  x.  A
)
1817eqcomi 2193 . . . . 5  |-  ( M  x.  A )  =  E
1918deceq1i 9425 . . . 4  |- ; ( M  x.  A
) 0  = ; E 0
2014, 16, 193eqtri 2214 . . 3  |-  ( M  x.  (; 1 0  x.  A
) )  = ; E 0
21 decmul10add.5 . . . 4  |-  F  =  ( M  x.  B
)
2221eqcomi 2193 . . 3  |-  ( M  x.  B )  =  F
2320, 22oveq12i 5912 . 2  |-  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )  =  (; E 0  +  F
)
242, 11, 233eqtri 2214 1  |-  ( M  x. ; A B )  =  (; E
0  +  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160  (class class class)co 5900   0cc0 7846   1c1 7847    + caddc 7849    x. cmul 7851   NN0cn0 9211  ;cdc 9419
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 4139  ax-pow 4195  ax-pr 4230  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957
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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-sub 8165  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-5 9016  df-6 9017  df-7 9018  df-8 9019  df-9 9020  df-n0 9212  df-dec 9420
This theorem is referenced by: (None)
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