Theorem nnmaxpwlemxy 12964

Description: Lemma for nnmaxpw 12969. Another way of stating that decomposing a natural number into a power of a base and a number not divisible by that base is unique. (Contributed by Jim Kingdon, 16-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)

Hypotheses

Ref	Expression
nnmaxpwlemxy.x	|- ( ph -> X e. NN )
nnmaxpwlemxy.b	|- ( ph -> B e. ( ZZ>= ` 2 ) )
nnmaxpwlemxy.bx	|- ( ph -> -. B || X )
nnmaxpwlemxy.y	|- ( ph -> Y e. NN0 )
nnmaxpwlemxy.ayx	|- ( ph -> A = ( ( B ^ Y ) x. X ) )

Assertion

Ref	Expression
nnmaxpwlemxy	|- ( ph -> ( X = ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) )

Distinct variable groups:   z, A   z, B   z, Y

Allowed substitution hints:   ph( z)   X( z)

Proof of Theorem nnmaxpwlemxy

Step	Hyp	Ref	Expression
1		nnmaxpwlemxy.b	. . . . . 6 |- ( ph -> B e. ( ZZ>= ` 2 ) )
2		eluz2nn 9975	. . . . . 6 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
3	1, 2	syl 14	. . . . 5 |- ( ph -> B e. NN )
4		nnmaxpwlemxy.x	. . . . . . . . 9 |- ( ph -> X e. NN )
5	4	nnzd 9771	. . . . . . . 8 |- ( ph -> X e. ZZ )
6		nnmaxpwlemxy.y	. . . . . . . . . 10 |- ( ph -> Y e. NN0 )
7	3, 6	nnexpcld 11146	. . . . . . . . 9 |- ( ph -> ( B ^ Y ) e. NN )
8	7	nnzd 9771	. . . . . . . 8 |- ( ph -> ( B ^ Y ) e. ZZ )
9		nnmaxpwlemxy.ayx	. . . . . . . . . 10 |- ( ph -> A = ( ( B ^ Y ) x. X ) )
10	7, 4	nnmulcld 9355	. . . . . . . . . 10 |- ( ph -> ( ( B ^ Y ) x. X ) e. NN )
11	9, 10	eqeltrd 2315	. . . . . . . . 9 |- ( ph -> A e. NN )
12	11	nnzd 9771	. . . . . . . 8 |- ( ph -> A e. ZZ )
13	7	nncnd 9320	. . . . . . . . . 10 |- ( ph -> ( B ^ Y ) e. CC )
14	4	nncnd 9320	. . . . . . . . . 10 |- ( ph -> X e. CC )
15	13, 14	mulcomd 8347	. . . . . . . . 9 |- ( ph -> ( ( B ^ Y ) x. X ) = ( X x. ( B ^ Y ) ) )
16	9, 15	eqtr2d 2272	. . . . . . . 8 |- ( ph -> ( X x. ( B ^ Y ) ) = A )
17		dvds0lem 12584	. . . . . . . 8 |- ( ( ( X e. ZZ /\ ( B ^ Y ) e. ZZ /\ A e. ZZ ) /\ ( X x. ( B ^ Y ) ) = A ) -> ( B ^ Y ) || A )
18	5, 8, 12, 16, 17	syl31anc 1281	. . . . . . 7 |- ( ph -> ( B ^ Y ) || A )
19		nnmaxpwlemxy.bx	. . . . . . . . 9 |- ( ph -> -. B || X )
20	9	breq2d 4142	. . . . . . . . . 10 |- ( ph -> ( ( ( B ^ Y ) x. B ) || A <-> ( ( B ^ Y ) x. B ) || ( ( B ^ Y ) x. X ) ) )
21	3	nnzd 9771	. . . . . . . . . . 11 |- ( ph -> B e. ZZ )
22	7	nnne0d 9351	. . . . . . . . . . 11 |- ( ph -> ( B ^ Y ) =/= 0 )
23		dvdscmulr 12603	. . . . . . . . . . 11 |- ( ( B e. ZZ /\ X e. ZZ /\ ( ( B ^ Y ) e. ZZ /\ ( B ^ Y ) =/= 0 ) ) -> ( ( ( B ^ Y ) x. B ) || ( ( B ^ Y ) x. X ) <-> B || X ) )
24	21, 5, 8, 22, 23	syl112anc 1282	. . . . . . . . . 10 |- ( ph -> ( ( ( B ^ Y ) x. B ) || ( ( B ^ Y ) x. X ) <-> B || X ) )
25	20, 24	bitrd 188	. . . . . . . . 9 |- ( ph -> ( ( ( B ^ Y ) x. B ) || A <-> B || X ) )
26	19, 25	mtbird 684	. . . . . . . 8 |- ( ph -> -. ( ( B ^ Y ) x. B ) || A )
27	3	nncnd 9320	. . . . . . . . . 10 |- ( ph -> B e. CC )
28	27, 6	expp1d 11125	. . . . . . . . 9 |- ( ph -> ( B ^ ( Y + 1 ) ) = ( ( B ^ Y ) x. B ) )
29	28	breq1d 4140	. . . . . . . 8 |- ( ph -> ( ( B ^ ( Y + 1 ) ) || A <-> ( ( B ^ Y ) x. B ) || A ) )
30	26, 29	mtbird 684	. . . . . . 7 |- ( ph -> -. ( B ^ ( Y + 1 ) ) || A )
31		pwbdvdseu 12963	. . . . . . . . 9 |- ( ( A e. NN /\ B e. ( ZZ>= ` 2 ) ) -> E! z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) )
32	11, 1, 31	syl2anc 415	. . . . . . . 8 |- ( ph -> E! z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) )
33		oveq2 6093	. . . . . . . . . . 11 |- ( z = Y -> ( B ^ z ) = ( B ^ Y ) )
34	33	breq1d 4140	. . . . . . . . . 10 |- ( z = Y -> ( ( B ^ z ) || A <-> ( B ^ Y ) || A ) )
35		oveq1 6092	. . . . . . . . . . . . 13 |- ( z = Y -> ( z + 1 ) = ( Y + 1 ) )
36	35	oveq2d 6101	. . . . . . . . . . . 12 |- ( z = Y -> ( B ^ ( z + 1 ) ) = ( B ^ ( Y + 1 ) ) )
37	36	breq1d 4140	. . . . . . . . . . 11 |- ( z = Y -> ( ( B ^ ( z + 1 ) ) || A <-> ( B ^ ( Y + 1 ) ) || A ) )
38	37	notbid 677	. . . . . . . . . 10 |- ( z = Y -> ( -. ( B ^ ( z + 1 ) ) || A <-> -. ( B ^ ( Y + 1 ) ) || A ) )
39	34, 38	anbi12d 477	. . . . . . . . 9 |- ( z = Y -> ( ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) <-> ( ( B ^ Y ) || A /\ -. ( B ^ ( Y + 1 ) ) || A ) ) )
40	39	riota2 6062	. . . . . . . 8 |- ( ( Y e. NN0 /\ E! z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) -> ( ( ( B ^ Y ) || A /\ -. ( B ^ ( Y + 1 ) ) || A ) <-> ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) = Y ) )
41	6, 32, 40	syl2anc 415	. . . . . . 7 |- ( ph -> ( ( ( B ^ Y ) || A /\ -. ( B ^ ( Y + 1 ) ) || A ) <-> ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) = Y ) )
42	18, 30, 41	mpbi2and 956	. . . . . 6 |- ( ph -> ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) = Y )
43	42, 6	eqeltrd 2315	. . . . 5 |- ( ph -> ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) e. NN0 )
44	3, 43	nnexpcld 11146	. . . 4 |- ( ph -> ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) e. NN )
45	44	nncnd 9320	. . 3 |- ( ph -> ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) e. CC )
46	44	nnap0d 9352	. . 3 |- ( ph -> ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) # 0 )
47	42	eqcomd 2244	. . . . . 6 |- ( ph -> Y = ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) )
48	47	oveq2d 6101	. . . . 5 |- ( ph -> ( B ^ Y ) = ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) )
49	48	oveq1d 6100	. . . 4 |- ( ph -> ( ( B ^ Y ) x. X ) = ( ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) x. X ) )
50	9, 49	eqtr2d 2272	. . 3 |- ( ph -> ( ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) x. X ) = A )
51	45, 14, 46, 50	mvllmulapd 9174	. 2 |- ( ph -> X = ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) )
52	51, 47	jca 306	1 |- ( ph -> ( X = ( A / ( B ^ ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) ) /\ Y = ( iota_ z e. NN0 ( ( B ^ z ) || A /\ -. ( B ^ ( z + 1 ) ) || A ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   E!wreu 2530   class class class wbr 4130   ` cfv 5377   iota_crio 6037  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   / cdiv 9004   NNcn 9306   2c2 9357   NN0cn0 9567   ZZcz 9648   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:  nnmaxpwlemparts  12968