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Theorem decmul10add 9398
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 9333 . . 3  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
21oveq2i 5861 . 2  |-  ( M  x. ; A B )  =  ( M  x.  ( (; 1
0  x.  A )  +  B ) )
3 decmul10add.3 . . . 4  |-  M  e. 
NN0
43nn0cni 9134 . . 3  |-  M  e.  CC
5 10nn0 9347 . . . . 5  |- ; 1 0  e.  NN0
6 decmul10add.1 . . . . 5  |-  A  e. 
NN0
75, 6nn0mulcli 9160 . . . 4  |-  (; 1 0  x.  A
)  e.  NN0
87nn0cni 9134 . . 3  |-  (; 1 0  x.  A
)  e.  CC
9 decmul10add.2 . . . 4  |-  B  e. 
NN0
109nn0cni 9134 . . 3  |-  B  e.  CC
114, 8, 10adddii 7917 . 2  |-  ( M  x.  ( (; 1 0  x.  A
)  +  B ) )  =  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )
125nn0cni 9134 . . . . 5  |- ; 1 0  e.  CC
136nn0cni 9134 . . . . 5  |-  A  e.  CC
144, 12, 13mul12i 8052 . . . 4  |-  ( M  x.  (; 1 0  x.  A
) )  =  (; 1
0  x.  ( M  x.  A ) )
153, 6nn0mulcli 9160 . . . . 5  |-  ( M  x.  A )  e. 
NN0
1615dec0u 9350 . . . 4  |-  (; 1 0  x.  ( M  x.  A )
)  = ; ( M  x.  A
) 0
17 decmul10add.4 . . . . . 6  |-  E  =  ( M  x.  A
)
1817eqcomi 2174 . . . . 5  |-  ( M  x.  A )  =  E
1918deceq1i 9336 . . . 4  |- ; ( M  x.  A
) 0  = ; E 0
2014, 16, 193eqtri 2195 . . 3  |-  ( M  x.  (; 1 0  x.  A
) )  = ; E 0
21 decmul10add.5 . . . 4  |-  F  =  ( M  x.  B
)
2221eqcomi 2174 . . 3  |-  ( M  x.  B )  =  F
2320, 22oveq12i 5862 . 2  |-  ( ( M  x.  (; 1 0  x.  A
) )  +  ( M  x.  B ) )  =  (; E 0  +  F
)
242, 11, 233eqtri 2195 1  |-  ( M  x. ; A B )  =  (; E
0  +  F )
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141  (class class class)co 5850   0cc0 7761   1c1 7762    + caddc 7764    x. cmul 7766   NN0cn0 9122  ;cdc 9330
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-sub 8079  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-5 8927  df-6 8928  df-7 8929  df-8 8930  df-9 8931  df-n0 9123  df-dec 9331
This theorem is referenced by: (None)
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