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Theorem decsplit 13346
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 10178 . . . . . 6  |-  10  e.  NN0
21nn0cni 10165 . . . . 5  |-  10  e.  CC
3 decsplit0.1 . . . . . . 7  |-  A  e. 
NN0
43nn0cni 10165 . . . . . 6  |-  A  e.  CC
5 decsplit.4 . . . . . . 7  |-  M  e. 
NN0
6 expcl 11326 . . . . . . 7  |-  ( ( 10  e.  CC  /\  M  e.  NN0 )  -> 
( 10 ^ M
)  e.  CC )
72, 5, 6mp2an 654 . . . . . 6  |-  ( 10
^ M )  e.  CC
84, 7mulcli 9028 . . . . 5  |-  ( A  x.  ( 10 ^ M ) )  e.  CC
92, 8mulcli 9028 . . . 4  |-  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  e.  CC
10 decsplit.2 . . . . . 6  |-  B  e. 
NN0
111, 10nn0mulcli 10190 . . . . 5  |-  ( 10  x.  B )  e. 
NN0
1211nn0cni 10165 . . . 4  |-  ( 10  x.  B )  e.  CC
13 decsplit.3 . . . . 5  |-  D  e. 
NN0
1413nn0cni 10165 . . . 4  |-  D  e.  CC
159, 12, 14addassi 9031 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
1610nn0cni 10165 . . . . . 6  |-  B  e.  CC
172, 8, 16adddii 9033 . . . . 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 6031 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( 10  x.  C
)
2017, 19eqtr3i 2409 . . . 4  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  =  ( 10  x.  C
)
2120oveq1i 6030 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  C
)  +  D )
2215, 21eqtr3i 2409 . 2  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )  =  ( ( 10  x.  C )  +  D
)
23 decsplit.5 . . . . . 6  |-  ( M  +  1 )  =  N
247, 2mulcomi 9029 . . . . . 6  |-  ( ( 10 ^ M )  x.  10 )  =  ( 10  x.  ( 10 ^ M ) )
251, 5, 23, 24numexpp1 13341 . . . . 5  |-  ( 10
^ N )  =  ( 10  x.  ( 10 ^ M ) )
2625oveq2i 6031 . . . 4  |-  ( A  x.  ( 10 ^ N ) )  =  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )
274, 2, 7mul12i 9193 . . . 4  |-  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
2826, 27eqtri 2407 . . 3  |-  ( A  x.  ( 10 ^ N ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
29 df-dec 10315 . . 3  |- ; B D  =  ( ( 10  x.  B
)  +  D )
3028, 29oveq12i 6032 . 2  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
31 df-dec 10315 . 2  |- ; C D  =  ( ( 10  x.  C
)  +  D )
3222, 30, 313eqtr4i 2417 1  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  = ; C D
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928   10c10 9989   NN0cn0 10153  ;cdc 10314   ^cexp 11309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-seq 11251  df-exp 11310
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