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Theorem decsplit 13098
Description: Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.)
Hypotheses
Ref Expression
decsplit0.1  |-  A  e. 
NN0
decsplit.2  |-  B  e. 
NN0
decsplit.3  |-  D  e. 
NN0
decsplit.4  |-  M  e. 
NN0
decsplit.5  |-  ( M  +  1 )  =  N
decsplit.6  |-  ( ( A  x.  ( 10
^ M ) )  +  B )  =  C
Assertion
Ref Expression
decsplit  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  = ; C D

Proof of Theorem decsplit
StepHypRef Expression
1 10nn0 9990 . . . . . 6  |-  10  e.  NN0
21nn0cni 9977 . . . . 5  |-  10  e.  CC
3 decsplit0.1 . . . . . . 7  |-  A  e. 
NN0
43nn0cni 9977 . . . . . 6  |-  A  e.  CC
5 decsplit.4 . . . . . . 7  |-  M  e. 
NN0
6 expcl 11121 . . . . . . 7  |-  ( ( 10  e.  CC  /\  M  e.  NN0 )  -> 
( 10 ^ M
)  e.  CC )
72, 5, 6mp2an 653 . . . . . 6  |-  ( 10
^ M )  e.  CC
84, 7mulcli 8842 . . . . 5  |-  ( A  x.  ( 10 ^ M ) )  e.  CC
92, 8mulcli 8842 . . . 4  |-  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  e.  CC
10 decsplit.2 . . . . . 6  |-  B  e. 
NN0
111, 10nn0mulcli 10002 . . . . 5  |-  ( 10  x.  B )  e. 
NN0
1211nn0cni 9977 . . . 4  |-  ( 10  x.  B )  e.  CC
13 decsplit.3 . . . . 5  |-  D  e. 
NN0
1413nn0cni 9977 . . . 4  |-  D  e.  CC
159, 12, 14addassi 8845 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
1610nn0cni 9977 . . . . . 6  |-  B  e.  CC
172, 8, 16adddii 8847 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )
18 decsplit.6 . . . . . 6  |-  ( ( A  x.  ( 10
^ M ) )  +  B )  =  C
1918oveq2i 5869 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( 10  x.  C
)
2017, 19eqtr3i 2305 . . . 4  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  =  ( 10  x.  C
)
2120oveq1i 5868 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  C
)  +  D )
2215, 21eqtr3i 2305 . 2  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )  =  ( ( 10  x.  C )  +  D
)
23 decsplit.5 . . . . . 6  |-  ( M  +  1 )  =  N
247, 2mulcomi 8843 . . . . . 6  |-  ( ( 10 ^ M )  x.  10 )  =  ( 10  x.  ( 10 ^ M ) )
251, 5, 23, 24numexpp1 13093 . . . . 5  |-  ( 10
^ N )  =  ( 10  x.  ( 10 ^ M ) )
2625oveq2i 5869 . . . 4  |-  ( A  x.  ( 10 ^ N ) )  =  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )
274, 2, 7mul12i 9007 . . . 4  |-  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
2826, 27eqtri 2303 . . 3  |-  ( A  x.  ( 10 ^ N ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
29 df-dec 10125 . . 3  |- ; B D  =  ( ( 10  x.  B
)  +  D )
3028, 29oveq12i 5870 . 2  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
31 df-dec 10125 . 2  |- ; C D  =  ( ( 10  x.  C
)  +  D )
3222, 30, 313eqtr4i 2313 1  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  = ; C D
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742   10c10 9803   NN0cn0 9965  ;cdc 10124   ^cexp 11104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-seq 11047  df-exp 11105
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