Theorem dignn0fr 42395

Description: The digits of the fractional part of a nonnegative integer are 0. (Contributed by AV, 23-May-2020.)

Assertion

Ref	Expression
dignn0fr	|- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( K (digit ` B ) N ) = 0 )

Proof of Theorem dignn0fr

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( B e. NN -> B e. NN )
2		eldifi 3732	. . 3 |- ( K e. ( ZZ \ NN0 ) -> K e. ZZ )
3		nn0re 11301	. . . 4 |- ( N e. NN0 -> N e. RR )
4		nn0ge0 11318	. . . 4 |- ( N e. NN0 -> 0 <_ N )
5		elrege0 12278	. . . 4 |- ( N e. ( 0 [,) +oo ) <-> ( N e. RR /\ 0 <_ N ) )
6	3, 4, 5	sylanbrc 698	. . 3 |- ( N e. NN0 -> N e. ( 0 [,) +oo ) )
7		digval 42392	. . 3 |- ( ( B e. NN /\ K e. ZZ /\ N e. ( 0 [,) +oo ) ) -> ( K (digit ` B ) N ) = ( ( |_ ` ( ( B ^ -u K ) x. N ) ) mod B ) )
8	1, 2, 6, 7	syl3an 1368	. 2 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( K (digit ` B ) N ) = ( ( |_ ` ( ( B ^ -u K ) x. N ) ) mod B ) )
9		nnz 11399	. . . . . . . 8 |- ( B e. NN -> B e. ZZ )
10		eldif 3584	. . . . . . . . . 10 |- ( K e. ( ZZ \ NN0 ) <-> ( K e. ZZ /\ -. K e. NN0 ) )
11		znnn0nn 11489	. . . . . . . . . 10 |- ( ( K e. ZZ /\ -. K e. NN0 ) -> -u K e. NN )
12	10, 11	sylbi 207	. . . . . . . . 9 |- ( K e. ( ZZ \ NN0 ) -> -u K e. NN )
13	12	nnnn0d 11351	. . . . . . . 8 |- ( K e. ( ZZ \ NN0 ) -> -u K e. NN0 )
14		zexpcl 12875	. . . . . . . 8 |- ( ( B e. ZZ /\ -u K e. NN0 ) -> ( B ^ -u K ) e. ZZ )
15	9, 13, 14	syl2an 494	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) ) -> ( B ^ -u K ) e. ZZ )
16	15	3adant3 1081	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B ^ -u K ) e. ZZ )
17		nn0z 11400	. . . . . . 7 |- ( N e. NN0 -> N e. ZZ )
18	17	3ad2ant3 1084	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> N e. ZZ )
19	16, 18	zmulcld 11488	. . . . 5 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( B ^ -u K ) x. N ) e. ZZ )
20		flid 12609	. . . . 5 |- ( ( ( B ^ -u K ) x. N ) e. ZZ -> ( |_ ` ( ( B ^ -u K ) x. N ) ) = ( ( B ^ -u K ) x. N ) )
21	19, 20	syl 17	. . . 4 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( |_ ` ( ( B ^ -u K ) x. N ) ) = ( ( B ^ -u K ) x. N ) )
22	21	oveq1d 6665	. . 3 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( |_ ` ( ( B ^ -u K ) x. N ) ) mod B ) = ( ( ( B ^ -u K ) x. N ) mod B ) )
23		nnre 11027	. . . . . . . . . 10 |- ( B e. NN -> B e. RR )
24		reexpcl 12877	. . . . . . . . . 10 |- ( ( B e. RR /\ -u K e. NN0 ) -> ( B ^ -u K ) e. RR )
25	23, 13, 24	syl2an 494	. . . . . . . . 9 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) ) -> ( B ^ -u K ) e. RR )
26	25	recnd 10068	. . . . . . . 8 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) ) -> ( B ^ -u K ) e. CC )
27	26	3adant3 1081	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B ^ -u K ) e. CC )
28		nn0cn 11302	. . . . . . . 8 |- ( N e. NN0 -> N e. CC )
29	28	3ad2ant3 1084	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> N e. CC )
30		nncn 11028	. . . . . . . . 9 |- ( B e. NN -> B e. CC )
31		nnne0 11053	. . . . . . . . 9 |- ( B e. NN -> B =/= 0 )
32	30, 31	jca 554	. . . . . . . 8 |- ( B e. NN -> ( B e. CC /\ B =/= 0 ) )
33	32	3ad2ant1 1082	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B e. CC /\ B =/= 0 ) )
34		div23 10704	. . . . . . 7 |- ( ( ( B ^ -u K ) e. CC /\ N e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( ( B ^ -u K ) x. N ) / B ) = ( ( ( B ^ -u K ) / B ) x. N ) )
35	27, 29, 33, 34	syl3anc 1326	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( B ^ -u K ) x. N ) / B ) = ( ( ( B ^ -u K ) / B ) x. N ) )
36	30	3ad2ant1 1082	. . . . . . . . 9 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> B e. CC )
37	31	3ad2ant1 1082	. . . . . . . . 9 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> B =/= 0 )
38	12	nnzd 11481	. . . . . . . . . 10 |- ( K e. ( ZZ \ NN0 ) -> -u K e. ZZ )
39	38	3ad2ant2 1083	. . . . . . . . 9 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> -u K e. ZZ )
40	36, 37, 39	expm1d 13018	. . . . . . . 8 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B ^ ( -u K - 1 ) ) = ( ( B ^ -u K ) / B ) )
41	40	eqcomd 2628	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( B ^ -u K ) / B ) = ( B ^ ( -u K - 1 ) ) )
42	41	oveq1d 6665	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( B ^ -u K ) / B ) x. N ) = ( ( B ^ ( -u K - 1 ) ) x. N ) )
43	35, 42	eqtrd 2656	. . . . 5 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( B ^ -u K ) x. N ) / B ) = ( ( B ^ ( -u K - 1 ) ) x. N ) )
44		nnm1nn0 11334	. . . . . . . . 9 |- ( -u K e. NN -> ( -u K - 1 ) e. NN0 )
45	12, 44	syl 17	. . . . . . . 8 |- ( K e. ( ZZ \ NN0 ) -> ( -u K - 1 ) e. NN0 )
46		zexpcl 12875	. . . . . . . 8 |- ( ( B e. ZZ /\ ( -u K - 1 ) e. NN0 ) -> ( B ^ ( -u K - 1 ) ) e. ZZ )
47	9, 45, 46	syl2an 494	. . . . . . 7 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) ) -> ( B ^ ( -u K - 1 ) ) e. ZZ )
48	47	3adant3 1081	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B ^ ( -u K - 1 ) ) e. ZZ )
49	48, 18	zmulcld 11488	. . . . 5 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( B ^ ( -u K - 1 ) ) x. N ) e. ZZ )
50	43, 49	eqeltrd 2701	. . . 4 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( B ^ -u K ) x. N ) / B ) e. ZZ )
51	25	3adant3 1081	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( B ^ -u K ) e. RR )
52	3	3ad2ant3 1084	. . . . . 6 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> N e. RR )
53	51, 52	remulcld 10070	. . . . 5 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( B ^ -u K ) x. N ) e. RR )
54		nnrp 11842	. . . . . 6 |- ( B e. NN -> B e. RR+ )
55	54	3ad2ant1 1082	. . . . 5 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> B e. RR+ )
56		mod0 12675	. . . . 5 |- ( ( ( ( B ^ -u K ) x. N ) e. RR /\ B e. RR+ ) -> ( ( ( ( B ^ -u K ) x. N ) mod B ) = 0 <-> ( ( ( B ^ -u K ) x. N ) / B ) e. ZZ ) )
57	53, 55, 56	syl2anc 693	. . . 4 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( ( B ^ -u K ) x. N ) mod B ) = 0 <-> ( ( ( B ^ -u K ) x. N ) / B ) e. ZZ ) )
58	50, 57	mpbird 247	. . 3 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( ( B ^ -u K ) x. N ) mod B ) = 0 )
59	22, 58	eqtrd 2656	. 2 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( ( |_ ` ( ( B ^ -u K ) x. N ) ) mod B ) = 0 )
60	8, 59	eqtrd 2656	1 |- ( ( B e. NN /\ K e. ( ZZ \ NN0 ) /\ N e. NN0 ) -> ( K (digit ` B ) N ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   +oocpnf 10071   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   [,)cico 12177   |_cfl 12591   mod cmo 12668   ^cexp 12860  digitcdig 42389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-dig 42390

This theorem is referenced by:  dig1  42402