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Theorem decsplit 13094
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 9986 . . . . . 6  |-  10  e.  NN0
21nn0cni 9973 . . . . 5  |-  10  e.  CC
3 decsplit0.1 . . . . . . 7  |-  A  e. 
NN0
43nn0cni 9973 . . . . . 6  |-  A  e.  CC
5 decsplit.4 . . . . . . 7  |-  M  e. 
NN0
6 expcl 11117 . . . . . . 7  |-  ( ( 10  e.  CC  /\  M  e.  NN0 )  -> 
( 10 ^ M
)  e.  CC )
72, 5, 6mp2an 653 . . . . . 6  |-  ( 10
^ M )  e.  CC
84, 7mulcli 8838 . . . . 5  |-  ( A  x.  ( 10 ^ M ) )  e.  CC
92, 8mulcli 8838 . . . 4  |-  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  e.  CC
10 decsplit.2 . . . . . 6  |-  B  e. 
NN0
111, 10nn0mulcli 9998 . . . . 5  |-  ( 10  x.  B )  e. 
NN0
1211nn0cni 9973 . . . 4  |-  ( 10  x.  B )  e.  CC
13 decsplit.3 . . . . 5  |-  D  e. 
NN0
1413nn0cni 9973 . . . 4  |-  D  e.  CC
159, 12, 14addassi 8841 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
1610nn0cni 9973 . . . . . 6  |-  B  e.  CC
172, 8, 16adddii 8843 . . . . 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 5831 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( 10  x.  C
)
2017, 19eqtr3i 2306 . . . 4  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  =  ( 10  x.  C
)
2120oveq1i 5830 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  C
)  +  D )
2215, 21eqtr3i 2306 . 2  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )  =  ( ( 10  x.  C )  +  D
)
23 decsplit.5 . . . . . 6  |-  ( M  +  1 )  =  N
247, 2mulcomi 8839 . . . . . 6  |-  ( ( 10 ^ M )  x.  10 )  =  ( 10  x.  ( 10 ^ M ) )
251, 5, 23, 24numexpp1 13089 . . . . 5  |-  ( 10
^ N )  =  ( 10  x.  ( 10 ^ M ) )
2625oveq2i 5831 . . . 4  |-  ( A  x.  ( 10 ^ N ) )  =  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )
274, 2, 7mul12i 9003 . . . 4  |-  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
2826, 27eqtri 2304 . . 3  |-  ( A  x.  ( 10 ^ N ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
29 df-dec 10121 . . 3  |- ; B D  =  ( ( 10  x.  B
)  +  D )
3028, 29oveq12i 5832 . 2  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
31 df-dec 10121 . 2  |- ; C D  =  ( ( 10  x.  C
)  +  D )
3222, 30, 313eqtr4i 2314 1  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  = ; C D
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1685  (class class class)co 5820   CCcc 8731   1c1 8734    + caddc 8736    x. cmul 8738   10c10 9799   NN0cn0 9961  ;cdc 10120   ^cexp 11100
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-seq 11043  df-exp 11101
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