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Theorem decmul10add 9262
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 9197 . . 3  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
21oveq2i 5785 . 2  |-  ( M  x. ; A B )  =  ( M  x.  ( (; 1
0  x.  A )  +  B ) )
3 decmul10add.3 . . . 4  |-  M  e. 
NN0
43nn0cni 9001 . . 3  |-  M  e.  CC
5 10nn0 9211 . . . . 5  |- ; 1 0  e.  NN0
6 decmul10add.1 . . . . 5  |-  A  e. 
NN0
75, 6nn0mulcli 9027 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
87nn0cni 9001 . . 3  |-  (; 1 0  x.  A
)  e.  CC
9 decmul10add.2 . . . 4  |-  B  e. 
NN0
109nn0cni 9001 . . 3  |-  B  e.  CC
114, 8, 10adddii 7788 . 2  |-  ( M  x.  ( (; 1 0  x.  A
)  +  B ) )  =  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )
125nn0cni 9001 . . . . 5  |- ; 1 0  e.  CC
136nn0cni 9001 . . . . 5  |-  A  e.  CC
144, 12, 13mul12i 7920 . . . 4  |-  ( M  x.  (; 1 0  x.  A
) )  =  (; 1
0  x.  ( M  x.  A ) )
153, 6nn0mulcli 9027 . . . . 5  |-  ( M  x.  A )  e. 
NN0
1615dec0u 9214 . . . 4  |-  (; 1 0  x.  ( M  x.  A )
)  = ; ( M  x.  A
) 0
17 decmul10add.4 . . . . . 6  |-  E  =  ( M  x.  A
)
1817eqcomi 2143 . . . . 5  |-  ( M  x.  A )  =  E
1918deceq1i 9200 . . . 4  |- ; ( M  x.  A
) 0  = ; E 0
2014, 16, 193eqtri 2164 . . 3  |-  ( M  x.  (; 1 0  x.  A
) )  = ; E 0
21 decmul10add.5 . . . 4  |-  F  =  ( M  x.  B
)
2221eqcomi 2143 . . 3  |-  ( M  x.  B )  =  F
2320, 22oveq12i 5786 . 2  |-  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )  =  (; E 0  +  F
)
242, 11, 233eqtri 2164 1  |-  ( M  x. ; A B )  =  (; E
0  +  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480  (class class class)co 5774   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637   NN0cn0 8989  ;cdc 9194
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7947  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-5 8794  df-6 8795  df-7 8796  df-8 8797  df-9 8798  df-n0 8990  df-dec 9195
This theorem is referenced by: (None)
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