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Theorem digit2 11504
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 9999 . . . . . . . 8  |-  ( B  e.  NN  ->  B  e.  RR )
2 nnnn0 10220 . . . . . . . 8  |-  ( K  e.  NN  ->  K  e.  NN0 )
3 reexpcl 11390 . . . . . . . 8  |-  ( ( B  e.  RR  /\  K  e.  NN0 )  -> 
( B ^ K
)  e.  RR )
41, 2, 3syl2an 464 . . . . . . 7  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K
)  e.  RR )
5 remulcl 9067 . . . . . . 7  |-  ( ( ( B ^ K
)  e.  RR  /\  A  e.  RR )  ->  ( ( B ^ K )  x.  A
)  e.  RR )
64, 5sylan 458 . . . . . 6  |-  ( ( ( B  e.  NN  /\  K  e.  NN )  /\  A  e.  RR )  ->  ( ( B ^ K )  x.  A )  e.  RR )
763impa 1148 . . . . 5  |-  ( ( B  e.  NN  /\  K  e.  NN  /\  A  e.  RR )  ->  (
( B ^ K
)  x.  A )  e.  RR )
873comr 1161 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( B ^ K
)  x.  A )  e.  RR )
9 reflcl 11197 . . . 4  |-  ( ( ( B ^ K
)  x.  A )  e.  RR  ->  ( |_ `  ( ( B ^ K )  x.  A ) )  e.  RR )
108, 9syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( B ^ K )  x.  A ) )  e.  RR )
11 nnrp 10613 . . . 4  |-  ( B  e.  NN  ->  B  e.  RR+ )
12113ad2ant2 979 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  B  e.  RR+ )
13 modval 11244 . . 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 643 . 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 958 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  B  e.  NN )
16 fldiv 11233 . . . . . 6  |-  ( ( ( ( B ^ K )  x.  A
)  e.  RR  /\  B  e.  NN )  ->  ( |_ `  (
( |_ `  (
( B ^ K
)  x.  A ) )  /  B ) )  =  ( |_
`  ( ( ( B ^ K )  x.  A )  /  B ) ) )
178, 15, 16syl2anc 643 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( |_
`  ( ( B ^ K )  x.  A ) )  /  B ) )  =  ( |_ `  (
( ( B ^ K )  x.  A
)  /  B ) ) )
18 nncn 10000 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  e.  CC )
19 expcl 11391 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  K  e.  NN0 )  -> 
( B ^ K
)  e.  CC )
2018, 2, 19syl2an 464 . . . . . . . . 9  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K
)  e.  CC )
21203adant1 975 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B ^ K )  e.  CC )
22 recn 9072 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
23223ad2ant1 978 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  A  e.  CC )
24 nnne0 10024 . . . . . . . . . 10  |-  ( B  e.  NN  ->  B  =/=  0 )
2518, 24jca 519 . . . . . . . . 9  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B  =/=  0 ) )
26253ad2ant2 979 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B  e.  CC  /\  B  =/=  0 ) )
27 div23 9689 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( ( B ^ K )  x.  A
)  /  B )  =  ( ( ( B ^ K )  /  B )  x.  A ) )
29 nnz 10295 . . . . . . . . . 10  |-  ( K  e.  NN  ->  K  e.  ZZ )
30 expm1 11421 . . . . . . . . . . 11  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  K  e.  ZZ )  ->  ( B ^ ( K  - 
1 ) )  =  ( ( B ^ K )  /  B
) )
31303expa 1153 . . . . . . . . . 10  |-  ( ( ( B  e.  CC  /\  B  =/=  0 )  /\  K  e.  ZZ )  ->  ( B ^
( K  -  1 ) )  =  ( ( B ^ K
)  /  B ) )
3225, 29, 31syl2an 464 . . . . . . . . 9  |-  ( ( B  e.  NN  /\  K  e.  NN )  ->  ( B ^ ( K  -  1 ) )  =  ( ( B ^ K )  /  B ) )
33323adant1 975 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( B ^ ( K  - 
1 ) )  =  ( ( B ^ K )  /  B
) )
3433oveq1d 6088 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( B ^ ( K  -  1 ) )  x.  A )  =  ( ( ( B ^ K )  /  B )  x.  A ) )
3528, 34eqtr4d 2470 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  (
( ( B ^ K )  x.  A
)  /  B )  =  ( ( B ^ ( K  - 
1 ) )  x.  A ) )
3635fveq2d 5724 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( ( B ^ K )  x.  A )  /  B ) )  =  ( |_ `  (
( B ^ ( K  -  1 ) )  x.  A ) ) )
3717, 36eqtrd 2467 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN  /\  K  e.  NN )  ->  ( |_ `  ( ( |_
`  ( ( B ^ K )  x.  A ) )  /  B ) )  =  ( |_ `  (
( B ^ ( K  -  1 ) )  x.  A ) ) )
3837oveq2d 6089 . . 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 6089 . 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 2467 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    - cmin 9283    / cdiv 9669   NNcn 9992   NN0cn0 10213   ZZcz 10274   RR+crp 10604   |_cfl 11193    mod cmo 11242   ^cexp 11374
This theorem is referenced by:  digit1  11505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375
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