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Theorem decmul10add 9102
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 9037 . . 3  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
21oveq2i 5717 . 2  |-  ( M  x. ; A B )  =  ( M  x.  ( (; 1
0  x.  A )  +  B ) )
3 decmul10add.3 . . . 4  |-  M  e. 
NN0
43nn0cni 8841 . . 3  |-  M  e.  CC
5 10nn0 9051 . . . . 5  |- ; 1 0  e.  NN0
6 decmul10add.1 . . . . 5  |-  A  e. 
NN0
75, 6nn0mulcli 8867 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
87nn0cni 8841 . . 3  |-  (; 1 0  x.  A
)  e.  CC
9 decmul10add.2 . . . 4  |-  B  e. 
NN0
109nn0cni 8841 . . 3  |-  B  e.  CC
114, 8, 10adddii 7648 . 2  |-  ( M  x.  ( (; 1 0  x.  A
)  +  B ) )  =  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )
125nn0cni 8841 . . . . 5  |- ; 1 0  e.  CC
136nn0cni 8841 . . . . 5  |-  A  e.  CC
144, 12, 13mul12i 7779 . . . 4  |-  ( M  x.  (; 1 0  x.  A
) )  =  (; 1
0  x.  ( M  x.  A ) )
153, 6nn0mulcli 8867 . . . . 5  |-  ( M  x.  A )  e. 
NN0
1615dec0u 9054 . . . 4  |-  (; 1 0  x.  ( M  x.  A )
)  = ; ( M  x.  A
) 0
17 decmul10add.4 . . . . . 6  |-  E  =  ( M  x.  A
)
1817eqcomi 2104 . . . . 5  |-  ( M  x.  A )  =  E
1918deceq1i 9040 . . . 4  |- ; ( M  x.  A
) 0  = ; E 0
2014, 16, 193eqtri 2124 . . 3  |-  ( M  x.  (; 1 0  x.  A
) )  = ; E 0
21 decmul10add.5 . . . 4  |-  F  =  ( M  x.  B
)
2221eqcomi 2104 . . 3  |-  ( M  x.  B )  =  F
2320, 22oveq12i 5718 . 2  |-  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )  =  (; E 0  +  F
)
242, 11, 233eqtri 2124 1  |-  ( M  x. ; A B )  =  (; E
0  +  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1299    e. wcel 1448  (class class class)co 5706   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505   NN0cn0 8829  ;cdc 9034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sub 7806  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-5 8640  df-6 8641  df-7 8642  df-8 8643  df-9 8644  df-n0 8830  df-dec 9035
This theorem is referenced by: (None)
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