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Theorem decsplit 13347
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 10179 . . . . . 6  |-  10  e.  NN0
21nn0cni 10166 . . . . 5  |-  10  e.  CC
3 decsplit0.1 . . . . . . 7  |-  A  e. 
NN0
43nn0cni 10166 . . . . . 6  |-  A  e.  CC
5 decsplit.4 . . . . . . 7  |-  M  e. 
NN0
6 expcl 11327 . . . . . . 7  |-  ( ( 10  e.  CC  /\  M  e.  NN0 )  -> 
( 10 ^ M
)  e.  CC )
72, 5, 6mp2an 654 . . . . . 6  |-  ( 10
^ M )  e.  CC
84, 7mulcli 9029 . . . . 5  |-  ( A  x.  ( 10 ^ M ) )  e.  CC
92, 8mulcli 9029 . . . 4  |-  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  e.  CC
10 decsplit.2 . . . . . 6  |-  B  e. 
NN0
111, 10nn0mulcli 10191 . . . . 5  |-  ( 10  x.  B )  e. 
NN0
1211nn0cni 10166 . . . 4  |-  ( 10  x.  B )  e.  CC
13 decsplit.3 . . . . 5  |-  D  e. 
NN0
1413nn0cni 10166 . . . 4  |-  D  e.  CC
159, 12, 14addassi 9032 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
1610nn0cni 10166 . . . . . 6  |-  B  e.  CC
172, 8, 16adddii 9034 . . . . 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 6032 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( 10  x.  C
)
2017, 19eqtr3i 2410 . . . 4  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  =  ( 10  x.  C
)
2120oveq1i 6031 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  C
)  +  D )
2215, 21eqtr3i 2410 . 2  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )  =  ( ( 10  x.  C )  +  D
)
23 decsplit.5 . . . . . 6  |-  ( M  +  1 )  =  N
247, 2mulcomi 9030 . . . . . 6  |-  ( ( 10 ^ M )  x.  10 )  =  ( 10  x.  ( 10 ^ M ) )
251, 5, 23, 24numexpp1 13342 . . . . 5  |-  ( 10
^ N )  =  ( 10  x.  ( 10 ^ M ) )
2625oveq2i 6032 . . . 4  |-  ( A  x.  ( 10 ^ N ) )  =  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )
274, 2, 7mul12i 9194 . . . 4  |-  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
2826, 27eqtri 2408 . . 3  |-  ( A  x.  ( 10 ^ N ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
29 df-dec 10316 . . 3  |- ; B D  =  ( ( 10  x.  B
)  +  D )
3028, 29oveq12i 6033 . 2  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
31 df-dec 10316 . 2  |- ; C D  =  ( ( 10  x.  C
)  +  D )
3222, 30, 313eqtr4i 2418 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 6021   CCcc 8922   1c1 8925    + caddc 8927    x. cmul 8929   10c10 9990   NN0cn0 10154  ;cdc 10315   ^cexp 11310
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-seq 11252  df-exp 11311
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