Theorem pwbdvdslemn 12960

Description: Lemma for pwbdvds 12961. If a natural number has some power of a base which does not divide it, there is a highest power of the base which does divide it. (Contributed by Jim Kingdon, 14-Nov-2021.) (Revised by Jim Kingdon, 17-Aug-2026.)

Hypotheses

Ref	Expression
pwbdvdslemn.n	|- ( ph -> N e. NN )
pwbdvdslemn.a	|- ( ph -> A e. NN )
pwbdvdslemn.b	|- ( ph -> B e. NN )
pwbdvdslemn.dvds	|- ( ph -> -. ( B ^ A ) || N )

Assertion

Ref	Expression
pwbdvdslemn	|- ( ph -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )

Distinct variable groups:   m, N   B, m   ph, m

Allowed substitution hint:   A( m)

Proof of Theorem pwbdvdslemn

Dummy variables w k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwbdvdslemn.dvds	. 2 |- ( ph -> -. ( B ^ A ) || N )
2		pwbdvdslemn.a	. . . 4 |- ( ph -> A e. NN )
3	2	adantr 276	. . 3 |- ( ( ph /\ -. ( B ^ A ) || N ) -> A e. NN )
4		oveq2 6093	. . . . . . . 8 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
5	4	breq1d 4140	. . . . . . 7 |- ( w = 1 -> ( ( B ^ w ) || N <-> ( B ^ 1 ) || N ) )
6	5	notbid 677	. . . . . 6 |- ( w = 1 -> ( -. ( B ^ w ) || N <-> -. ( B ^ 1 ) || N ) )
7	6	anbi2d 468	. . . . 5 |- ( w = 1 -> ( ( ph /\ -. ( B ^ w ) || N ) <-> ( ph /\ -. ( B ^ 1 ) || N ) ) )
8	7	imbi1d 231	. . . 4 |- ( w = 1 -> ( ( ( ph /\ -. ( B ^ w ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) <-> ( ( ph /\ -. ( B ^ 1 ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) )
9		oveq2 6093	. . . . . . . 8 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
10	9	breq1d 4140	. . . . . . 7 |- ( w = k -> ( ( B ^ w ) || N <-> ( B ^ k ) || N ) )
11	10	notbid 677	. . . . . 6 |- ( w = k -> ( -. ( B ^ w ) || N <-> -. ( B ^ k ) || N ) )
12	11	anbi2d 468	. . . . 5 |- ( w = k -> ( ( ph /\ -. ( B ^ w ) || N ) <-> ( ph /\ -. ( B ^ k ) || N ) ) )
13	12	imbi1d 231	. . . 4 |- ( w = k -> ( ( ( ph /\ -. ( B ^ w ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) <-> ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) )
14		oveq2 6093	. . . . . . . 8 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
15	14	breq1d 4140	. . . . . . 7 |- ( w = ( k + 1 ) -> ( ( B ^ w ) || N <-> ( B ^ ( k + 1 ) ) || N ) )
16	15	notbid 677	. . . . . 6 |- ( w = ( k + 1 ) -> ( -. ( B ^ w ) || N <-> -. ( B ^ ( k + 1 ) ) || N ) )
17	16	anbi2d 468	. . . . 5 |- ( w = ( k + 1 ) -> ( ( ph /\ -. ( B ^ w ) || N ) <-> ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) )
18	17	imbi1d 231	. . . 4 |- ( w = ( k + 1 ) -> ( ( ( ph /\ -. ( B ^ w ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) <-> ( ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) )
19		oveq2 6093	. . . . . . . 8 |- ( w = A -> ( B ^ w ) = ( B ^ A ) )
20	19	breq1d 4140	. . . . . . 7 |- ( w = A -> ( ( B ^ w ) || N <-> ( B ^ A ) || N ) )
21	20	notbid 677	. . . . . 6 |- ( w = A -> ( -. ( B ^ w ) || N <-> -. ( B ^ A ) || N ) )
22	21	anbi2d 468	. . . . 5 |- ( w = A -> ( ( ph /\ -. ( B ^ w ) || N ) <-> ( ph /\ -. ( B ^ A ) || N ) ) )
23	22	imbi1d 231	. . . 4 |- ( w = A -> ( ( ( ph /\ -. ( B ^ w ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) <-> ( ( ph /\ -. ( B ^ A ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) )
24		oveq2 6093	. . . . . . 7 |- ( m = 0 -> ( B ^ m ) = ( B ^ 0 ) )
25	24	breq1d 4140	. . . . . 6 |- ( m = 0 -> ( ( B ^ m ) || N <-> ( B ^ 0 ) || N ) )
26		oveq1 6092	. . . . . . . . 9 |- ( m = 0 -> ( m + 1 ) = ( 0 + 1 ) )
27	26	oveq2d 6101	. . . . . . . 8 |- ( m = 0 -> ( B ^ ( m + 1 ) ) = ( B ^ ( 0 + 1 ) ) )
28	27	breq1d 4140	. . . . . . 7 |- ( m = 0 -> ( ( B ^ ( m + 1 ) ) || N <-> ( B ^ ( 0 + 1 ) ) || N ) )
29	28	notbid 677	. . . . . 6 |- ( m = 0 -> ( -. ( B ^ ( m + 1 ) ) || N <-> -. ( B ^ ( 0 + 1 ) ) || N ) )
30	25, 29	anbi12d 477	. . . . 5 |- ( m = 0 -> ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) <-> ( ( B ^ 0 ) || N /\ -. ( B ^ ( 0 + 1 ) ) || N ) ) )
31		0nn0 9582	. . . . . 6 |- 0 e. NN0
32	31	a1i 9	. . . . 5 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> 0 e. NN0 )
33		pwbdvdslemn.b	. . . . . . . . . 10 |- ( ph -> B e. NN )
34	33	nncnd 9320	. . . . . . . . 9 |- ( ph -> B e. CC )
35	34	adantr 276	. . . . . . . 8 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> B e. CC )
36	35	exp0d 11118	. . . . . . 7 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> ( B ^ 0 ) = 1 )
37		pwbdvdslemn.n	. . . . . . . . . 10 |- ( ph -> N e. NN )
38	37	adantr 276	. . . . . . . . 9 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> N e. NN )
39	38	nnzd 9771	. . . . . . . 8 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> N e. ZZ )
40		1dvds 12588	. . . . . . . 8 |- ( N e. ZZ -> 1 || N )
41	39, 40	syl 14	. . . . . . 7 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> 1 || N )
42	36, 41	eqbrtrd 4152	. . . . . 6 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> ( B ^ 0 ) || N )
43		simpr 110	. . . . . . 7 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> -. ( B ^ 1 ) || N )
44		0p1e1 9420	. . . . . . . . 9 |- ( 0 + 1 ) = 1
45	44	oveq2i 6096	. . . . . . . 8 |- ( B ^ ( 0 + 1 ) ) = ( B ^ 1 )
46	45	breq1i 4137	. . . . . . 7 |- ( ( B ^ ( 0 + 1 ) ) || N <-> ( B ^ 1 ) || N )
47	43, 46	sylnibr 688	. . . . . 6 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> -. ( B ^ ( 0 + 1 ) ) || N )
48	42, 47	jca 306	. . . . 5 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> ( ( B ^ 0 ) || N /\ -. ( B ^ ( 0 + 1 ) ) || N ) )
49	30, 32, 48	rspcedvdw 2936	. . . 4 |- ( ( ph /\ -. ( B ^ 1 ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
50		oveq2 6093	. . . . . . . . . 10 |- ( m = k -> ( B ^ m ) = ( B ^ k ) )
51	50	breq1d 4140	. . . . . . . . 9 |- ( m = k -> ( ( B ^ m ) || N <-> ( B ^ k ) || N ) )
52		oveq1 6092	. . . . . . . . . . . 12 |- ( m = k -> ( m + 1 ) = ( k + 1 ) )
53	52	oveq2d 6101	. . . . . . . . . . 11 |- ( m = k -> ( B ^ ( m + 1 ) ) = ( B ^ ( k + 1 ) ) )
54	53	breq1d 4140	. . . . . . . . . 10 |- ( m = k -> ( ( B ^ ( m + 1 ) ) || N <-> ( B ^ ( k + 1 ) ) || N ) )
55	54	notbid 677	. . . . . . . . 9 |- ( m = k -> ( -. ( B ^ ( m + 1 ) ) || N <-> -. ( B ^ ( k + 1 ) ) || N ) )
56	51, 55	anbi12d 477	. . . . . . . 8 |- ( m = k -> ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) <-> ( ( B ^ k ) || N /\ -. ( B ^ ( k + 1 ) ) || N ) ) )
57		simpll 531	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> k e. NN )
58	57	nnnn0d 9624	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> k e. NN0 )
59		simpr 110	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> ( B ^ k ) || N )
60		simplrr 542	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> -. ( B ^ ( k + 1 ) ) || N )
61	59, 60	jca 306	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> ( ( B ^ k ) || N /\ -. ( B ^ ( k + 1 ) ) || N ) )
62	56, 58, 61	rspcedvdw 2936	. . . . . . 7 |- ( ( ( k e. NN /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
63	62	adantllr 485	. . . . . 6 |- ( ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
64		simprl 535	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> ph )
65	64	anim1i 340	. . . . . . 7 |- ( ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ -. ( B ^ k ) || N ) -> ( ph /\ -. ( B ^ k ) || N ) )
66		simpllr 540	. . . . . . 7 |- ( ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ -. ( B ^ k ) || N ) -> ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) )
67	65, 66	mpd 13	. . . . . 6 |- ( ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
68	33	ad2antrl 494	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> B e. NN )
69		nnnn0 9574	. . . . . . . . . 10 |- ( k e. NN -> k e. NN0 )
70	69	ad2antrr 492	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> k e. NN0 )
71	68, 70	nnexpcld 11146	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> ( B ^ k ) e. NN )
72	37	ad2antrl 494	. . . . . . . . 9 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> N e. NN )
73	72	nnzd 9771	. . . . . . . 8 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> N e. ZZ )
74		dvdsdc 12581	. . . . . . . 8 |- ( ( ( B ^ k ) e. NN /\ N e. ZZ ) -> DECID ( B ^ k ) || N )
75	71, 73, 74	syl2anc 415	. . . . . . 7 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> DECID ( B ^ k ) || N )
76		exmiddc 848	. . . . . . 7 |- (DECID ( B ^ k ) || N -> ( ( B ^ k ) || N \/ -. ( B ^ k ) || N ) )
77	75, 76	syl 14	. . . . . 6 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> ( ( B ^ k ) || N \/ -. ( B ^ k ) || N ) )
78	63, 67, 77	mpjaodan 810	. . . . 5 |- ( ( ( k e. NN /\ ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) /\ ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
79	78	exp31 364	. . . 4 |- ( k e. NN -> ( ( ( ph /\ -. ( B ^ k ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) -> ( ( ph /\ -. ( B ^ ( k + 1 ) ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) ) )
80	8, 13, 18, 23, 49, 79	nnind 9322	. . 3 |- ( A e. NN -> ( ( ph /\ -. ( B ^ A ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) )
81	3, 80	mpcom 36	. 2 |- ( ( ph /\ -. ( B ^ A ) || N ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
82	1, 81	mpdan 425	1 |- ( ph -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   NNcn 9306   NN0cn0 9567   ZZcz 9648   ^cexp 10988   || cdvds 12570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-dvds 12571

This theorem is used by:  pwbdvds  12961