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Theorem decmul10add 8626
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 8561 . . 3  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
21oveq2i 5554 . 2  |-  ( M  x. ; A B )  =  ( M  x.  ( (; 1
0  x.  A )  +  B ) )
3 decmul10add.3 . . . 4  |-  M  e. 
NN0
43nn0cni 8367 . . 3  |-  M  e.  CC
5 10nn0 8575 . . . . 5  |- ; 1 0  e.  NN0
6 decmul10add.1 . . . . 5  |-  A  e. 
NN0
75, 6nn0mulcli 8393 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
87nn0cni 8367 . . 3  |-  (; 1 0  x.  A
)  e.  CC
9 decmul10add.2 . . . 4  |-  B  e. 
NN0
109nn0cni 8367 . . 3  |-  B  e.  CC
114, 8, 10adddii 7191 . 2  |-  ( M  x.  ( (; 1 0  x.  A
)  +  B ) )  =  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )
125nn0cni 8367 . . . . 5  |- ; 1 0  e.  CC
136nn0cni 8367 . . . . 5  |-  A  e.  CC
144, 12, 13mul12i 7321 . . . 4  |-  ( M  x.  (; 1 0  x.  A
) )  =  (; 1
0  x.  ( M  x.  A ) )
153, 6nn0mulcli 8393 . . . . 5  |-  ( M  x.  A )  e. 
NN0
1615dec0u 8578 . . . 4  |-  (; 1 0  x.  ( M  x.  A )
)  = ; ( M  x.  A
) 0
17 decmul10add.4 . . . . . 6  |-  E  =  ( M  x.  A
)
1817eqcomi 2086 . . . . 5  |-  ( M  x.  A )  =  E
1918deceq1i 8564 . . . 4  |- ; ( M  x.  A
) 0  = ; E 0
2014, 16, 193eqtri 2106 . . 3  |-  ( M  x.  (; 1 0  x.  A
) )  = ; E 0
21 decmul10add.5 . . . 4  |-  F  =  ( M  x.  B
)
2221eqcomi 2086 . . 3  |-  ( M  x.  B )  =  F
2320, 22oveq12i 5555 . 2  |-  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )  =  (; E 0  +  F
)
242, 11, 233eqtri 2106 1  |-  ( M  x. ; A B )  =  (; E
0  +  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    e. wcel 1434  (class class class)co 5543   0cc0 7043   1c1 7044    + caddc 7046    x. cmul 7048   NN0cn0 8355  ;cdc 8558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-5 8168  df-6 8169  df-7 8170  df-8 8171  df-9 8172  df-n0 8356  df-dec 8559
This theorem is referenced by: (None)
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