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Theorem digit2 11234
Description: Two ways to express the  K th digit in the decimal (when base  B  =  10) expansion of a number  A.  K  =  1 corresponds to the first digit after the decimal point. (Contributed by NM, 25-Dec-2008.)
Assertion
Ref Expression
digit2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( |_ `  (
( B ^ K
)  x.  A ) )  mod  B )  =  ( ( |_
`  ( ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( B ^ ( K  - 
1 ) )  x.  A ) ) ) ) )

Proof of Theorem digit2
StepHypRef Expression
1 nnre 9753 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  RR )
2 nnnn0 9972 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 reexpcl 11120 . . . . . . . 8  |-  ( ( B  e.  RR  /\  K  e.  NN0 )  -> 
( B ^ K
)  e.  RR )
41, 2, 3syl2an 463 . . . . . . 7  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K
)  e.  RR )
5 remulcl 8822 . . . . . . 7  |-  ( ( ( B ^ K
)  e.  RR  /\  A  e.  RR )  ->  ( ( B ^ K )  x.  A
)  e.  RR )
64, 5sylan 457 . . . . . 6  |-  ( ( ( B  e.  NN  /\  K  e.  NN )  /\  A  e.  RR )  ->  ( ( B ^ K )  x.  A )  e.  RR )
763impa 1146 . . . . 5  |-  ( ( B  e.  NN  /\  K  e.  NN  /\  A  e.  RR )  ->  (
( B ^ K
)  x.  A )  e.  RR )
873comr 1159 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( B ^ K
)  x.  A )  e.  RR )
9 reflcl 10928 . . . 4  |-  ( ( ( B ^ K
)  x.  A )  e.  RR  ->  ( |_ `  ( ( B ^ K )  x.  A ) )  e.  RR )
108, 9syl 15 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( B ^ K )  x.  A ) )  e.  RR )
11 nnrp 10363 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR+ )
12113ad2ant2 977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  B  e.  RR+ )
13 modval 10975 . . 3  |-  ( ( ( |_ `  (
( B ^ K
)  x.  A ) )  e.  RR  /\  B  e.  RR+ )  -> 
( ( |_ `  ( ( B ^ K )  x.  A
) )  mod  B
)  =  ( ( |_ `  ( ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( |_ `  ( ( B ^ K )  x.  A ) )  /  B ) ) ) ) )
1410, 12, 13syl2anc 642 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( |_ `  (
( B ^ K
)  x.  A ) )  mod  B )  =  ( ( |_
`  ( ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( |_
`  ( ( B ^ K )  x.  A ) )  /  B ) ) ) ) )
15 simp2 956 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  B  e.  NN )
16 fldiv 10964 . . . . . 6  |-  ( ( ( ( B ^ K )  x.  A
)  e.  RR  /\  B  e.  NN )  ->  ( |_ `  (
( |_ `  (
( B ^ K
)  x.  A ) )  /  B ) )  =  ( |_
`  ( ( ( B ^ K )  x.  A )  /  B ) ) )
178, 15, 16syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( |_
`  ( ( B ^ K )  x.  A ) )  /  B ) )  =  ( |_ `  (
( ( B ^ K )  x.  A
)  /  B ) ) )
18 nncn 9754 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  CC )
19 expcl 11121 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  K  e.  NN0 )  -> 
( B ^ K
)  e.  CC )
2018, 2, 19syl2an 463 . . . . . . . . 9  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K
)  e.  CC )
21203adant1 973 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K )  e.  CC )
22 recn 8827 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
23223ad2ant1 976 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  A  e.  CC )
24 nnne0 9778 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  =/=  0 )
2518, 24jca 518 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
26253ad2ant2 977 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
27 div23 9443 . . . . . . . 8  |-  ( ( ( B ^ K
)  e.  CC  /\  A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( ( B ^ K )  x.  A )  /  B
)  =  ( ( ( B ^ K
)  /  B )  x.  A ) )
2821, 23, 26, 27syl3anc 1182 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( ( B ^ K )  x.  A
)  /  B )  =  ( ( ( B ^ K )  /  B )  x.  A ) )
29 nnz 10045 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  ZZ )
30 expm1 11151 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  K  e.  ZZ )  ->  ( B ^ ( K  - 
1 ) )  =  ( ( B ^ K )  /  B
) )
31303expa 1151 . . . . . . . . . 10  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  K  e.  ZZ )  ->  ( B ^
( K  -  1 ) )  =  ( ( B ^ K
)  /  B ) )
3225, 29, 31syl2an 463 . . . . . . . . 9  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ ( K  -  1 ) )  =  ( ( B ^ K )  /  B ) )
33323adant1 973 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B ^ ( K  - 
1 ) )  =  ( ( B ^ K )  /  B
) )
3433oveq1d 5873 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( B ^ ( K  -  1 ) )  x.  A )  =  ( ( ( B ^ K )  /  B )  x.  A ) )
3528, 34eqtr4d 2318 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( ( B ^ K )  x.  A
)  /  B )  =  ( ( B ^ ( K  - 
1 ) )  x.  A ) )
3635fveq2d 5529 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( ( B ^ K )  x.  A )  /  B ) )  =  ( |_ `  (
( B ^ ( K  -  1 ) )  x.  A ) ) )
3717, 36eqtrd 2315 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( |_
`  ( ( B ^ K )  x.  A ) )  /  B ) )  =  ( |_ `  (
( B ^ ( K  -  1 ) )  x.  A ) ) )
3837oveq2d 5874 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B  x.  ( |_ `  ( ( |_ `  ( ( B ^ K )  x.  A
) )  /  B
) ) )  =  ( B  x.  ( |_ `  ( ( B ^ ( K  - 
1 ) )  x.  A ) ) ) )
3938oveq2d 5874 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( |_ `  (
( B ^ K
)  x.  A ) )  -  ( B  x.  ( |_ `  ( ( |_ `  ( ( B ^ K )  x.  A
) )  /  B
) ) ) )  =  ( ( |_
`  ( ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( B ^ ( K  - 
1 ) )  x.  A ) ) ) ) )
4014, 39eqtrd 2315 1  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( |_ `  (
( B ^ K
)  x.  A ) )  mod  B )  =  ( ( |_
`  ( ( B ^ K )  x.  A ) )  -  ( B  x.  ( |_ `  ( ( B ^ ( K  - 
1 ) )  x.  A ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037    / cdiv 9423   NNcn 9746   NN0cn0 9965   ZZcz 10024   RR+crp 10354   |_cfl 10924    mod cmo 10973   ^cexp 11104
This theorem is referenced by:  digit1  11235
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  ax-pre-sup 8815
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-rmo 2551  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-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105
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