MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  decsplit Structured version   Unicode version

Theorem decsplit 15018
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 10894 . . . . . 6  |-  10  e.  NN0
21nn0cni 10881 . . . . 5  |-  10  e.  CC
3 decsplit0.1 . . . . . . 7  |-  A  e. 
NN0
43nn0cni 10881 . . . . . 6  |-  A  e.  CC
5 decsplit.4 . . . . . . 7  |-  M  e. 
NN0
6 expcl 12287 . . . . . . 7  |-  ( ( 10  e.  CC  /\  M  e.  NN0 )  -> 
( 10 ^ M
)  e.  CC )
72, 5, 6mp2an 676 . . . . . 6  |-  ( 10
^ M )  e.  CC
84, 7mulcli 9647 . . . . 5  |-  ( A  x.  ( 10 ^ M ) )  e.  CC
92, 8mulcli 9647 . . . 4  |-  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  e.  CC
10 decsplit.2 . . . . . 6  |-  B  e. 
NN0
111, 10nn0mulcli 10908 . . . . 5  |-  ( 10  x.  B )  e. 
NN0
1211nn0cni 10881 . . . 4  |-  ( 10  x.  B )  e.  CC
13 decsplit.3 . . . . 5  |-  D  e. 
NN0
1413nn0cni 10881 . . . 4  |-  D  e.  CC
159, 12, 14addassi 9650 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
1610nn0cni 10881 . . . . . 6  |-  B  e.  CC
172, 8, 16adddii 9652 . . . . 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 6316 . . . . 5  |-  ( 10  x.  ( ( A  x.  ( 10 ^ M ) )  +  B ) )  =  ( 10  x.  C
)
2017, 19eqtr3i 2460 . . . 4  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  =  ( 10  x.  C
)
2120oveq1i 6315 . . 3  |-  ( ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( 10  x.  B ) )  +  D )  =  ( ( 10  x.  C
)  +  D )
2215, 21eqtr3i 2460 . 2  |-  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )  =  ( ( 10  x.  C )  +  D
)
23 decsplit.5 . . . . . 6  |-  ( M  +  1 )  =  N
247, 2mulcomi 9648 . . . . . 6  |-  ( ( 10 ^ M )  x.  10 )  =  ( 10  x.  ( 10 ^ M ) )
251, 5, 23, 24numexpp1 15013 . . . . 5  |-  ( 10
^ N )  =  ( 10  x.  ( 10 ^ M ) )
2625oveq2i 6316 . . . 4  |-  ( A  x.  ( 10 ^ N ) )  =  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )
274, 2, 7mul12i 9827 . . . 4  |-  ( A  x.  ( 10  x.  ( 10 ^ M ) ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
2826, 27eqtri 2458 . . 3  |-  ( A  x.  ( 10 ^ N ) )  =  ( 10  x.  ( A  x.  ( 10 ^ M ) ) )
29 df-dec 11052 . . 3  |- ; B D  =  ( ( 10  x.  B
)  +  D )
3028, 29oveq12i 6317 . 2  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  =  ( ( 10  x.  ( A  x.  ( 10 ^ M ) ) )  +  ( ( 10  x.  B )  +  D ) )
31 df-dec 11052 . 2  |- ; C D  =  ( ( 10  x.  C
)  +  D )
3222, 30, 313eqtr4i 2468 1  |-  ( ( A  x.  ( 10
^ N ) )  + ; B D )  = ; C D
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870  (class class class)co 6305   CCcc 9536   1c1 9539    + caddc 9541    x. cmul 9543   10c10 10667   NN0cn0 10869  ;cdc 11051   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-seq 12211  df-exp 12270
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator