Theorem pwbdvdseu 12963

Description: A natural number has a unique highest power of a base which divides it. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)

Assertion

Ref	Expression
pwbdvdseu	|- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E! m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )

Distinct variable groups:   m, N   B, m

Proof of Theorem pwbdvdseu

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwbdvds 12961	. 2 |- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
2		simplrl 541	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> m e. NN0 )
3	2	nn0red 9625	. . . . . 6 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> m e. RR )
4		simplrr 542	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> x e. NN0 )
5	4	nn0red 9625	. . . . . 6 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> x e. RR )
6		simplll 539	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> N e. NN )
7		eluz2nn 9975	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	7	ad3antlr 497	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> B e. NN )
9		simprll 543	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> ( B ^ m ) || N )
10		simprrr 546	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> -. ( B ^ ( x + 1 ) ) || N )
11	6, 2, 4, 8, 9, 10	pwbdvdseulemle 12962	. . . . . 6 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> m <_ x )
12		simprrl 545	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> ( B ^ x ) || N )
13		simprlr 544	. . . . . . 7 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> -. ( B ^ ( m + 1 ) ) || N )
14	6, 4, 2, 8, 12, 13	pwbdvdseulemle 12962	. . . . . 6 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> x <_ m )
15	3, 5, 11, 14	letrid 8443	. . . . 5 |- ( ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) /\ ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) ) -> m = x )
16	15	ex 115	. . . 4 |- ( ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) /\ ( m e. NN0 /\ x e. NN0 ) ) -> ( ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) -> m = x ) )
17	16	ralrimivva 2632	. . 3 |- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> A. m e. NN0 A. x e. NN0 ( ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) -> m = x ) )
18		oveq2 6093	. . . . . 6 |- ( m = x -> ( B ^ m ) = ( B ^ x ) )
19	18	breq1d 4140	. . . . 5 |- ( m = x -> ( ( B ^ m ) || N <-> ( B ^ x ) || N ) )
20		oveq1 6092	. . . . . . . 8 |- ( m = x -> ( m + 1 ) = ( x + 1 ) )
21	20	oveq2d 6101	. . . . . . 7 |- ( m = x -> ( B ^ ( m + 1 ) ) = ( B ^ ( x + 1 ) ) )
22	21	breq1d 4140	. . . . . 6 |- ( m = x -> ( ( B ^ ( m + 1 ) ) || N <-> ( B ^ ( x + 1 ) ) || N ) )
23	22	notbid 677	. . . . 5 |- ( m = x -> ( -. ( B ^ ( m + 1 ) ) || N <-> -. ( B ^ ( x + 1 ) ) || N ) )
24	19, 23	anbi12d 477	. . . 4 |- ( m = x -> ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) <-> ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) )
25	24	rmo4 3019	. . 3 |- ( E* m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) <-> A. m e. NN0 A. x e. NN0 ( ( ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ ( ( B ^ x ) || N /\ -. ( B ^ ( x + 1 ) ) || N ) ) -> m = x ) )
26	17, 25	sylibr 134	. 2 |- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E* m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )
27		reu5 2770	. 2 |- ( E! m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) <-> ( E. m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) /\ E* m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) ) )
28	1, 26, 27	sylanbrc 421	1 |- ( ( N e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E! m e. NN0 ( ( B ^ m ) || N /\ -. ( B ^ ( m + 1 ) ) || N ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2209   A.wral 2528   E.wrex 2529   E!wreu 2530   E*wrmo 2531   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182   NNcn 9306   2c2 9357   NN0cn0 9567   ZZ>=cuz 9930   ^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-2 9365  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-dvds 12571

This theorem is used by:  nnmaxpwlemxy  12964  nnmaxpwlemdvds  12965  nnmaxpwlemndvds  12966  nnmaxpwlemnfac  12967  nnmaxpwlemparts  12968  nnmaxpw  12969