Theorem zprmlogbaplem3 16217

Description: Lemma for zprmlogbap 16218. Decomposing a natural number into a power of a prime base and a factor not divisible by that prime. (Contributed by Jim Kingdon, 20-Aug-2026.)

Hypotheses

Ref	Expression
zprmlogbaplem3.j	|- J = { z e. NN | -. B || z }
zprmlogbaplem3.f	|- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )

Assertion

Ref	Expression
zprmlogbaplem3	|- ( ( X e. NN /\ B e. Prime ) -> E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) )

Distinct variable groups:   B, a, m   x, B, y, z   F, a, m   x, F, y, z   x, J, y   X, a, m   x, X, y, z

Allowed substitution hints:   J( z, m, a)

Proof of Theorem zprmlogbaplem3

Step	Hyp	Ref	Expression
1		prmuz2 12935	. . . . . . . . 9 |- ( B e. Prime -> B e. ( ZZ>= ` 2 ) )
2		zprmlogbaplem3.j	. . . . . . . . . 10 |- J = { z e. NN | -. B || z }
3		zprmlogbaplem3.f	. . . . . . . . . 10 |- F = ( x e. J , y e. NN0 |-> ( ( B ^ y ) x. x ) )
4	2, 3	nnmaxpw 12978	. . . . . . . . 9 |- ( B e. ( ZZ>= ` 2 ) -> F : ( J X. NN0 ) -1-1-onto-> NN )
5	1, 4	syl 14	. . . . . . . 8 |- ( B e. Prime -> F : ( J X. NN0 ) -1-1-onto-> NN )
6		simpl 109	. . . . . . . 8 |- ( ( X e. NN /\ B e. Prime ) -> X e. NN )
7		f1ocnvdm 5990	. . . . . . . 8 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ X e. NN ) -> ( `' F ` X ) e. ( J X. NN0 ) )
8	5, 6, 7	syl2an2 602	. . . . . . 7 |- ( ( X e. NN /\ B e. Prime ) -> ( `' F ` X ) e. ( J X. NN0 ) )
9		elxp6 6407	. . . . . . 7 |- ( ( `' F ` X ) e. ( J X. NN0 ) <-> ( ( `' F ` X ) = <. ( 1st ` ( `' F ` X ) ) , ( 2nd ` ( `' F ` X ) ) >. /\ ( ( 1st ` ( `' F ` X ) ) e. J /\ ( 2nd ` ( `' F ` X ) ) e. NN0 ) ) )
10	8, 9	sylib 122	. . . . . 6 |- ( ( X e. NN /\ B e. Prime ) -> ( ( `' F ` X ) = <. ( 1st ` ( `' F ` X ) ) , ( 2nd ` ( `' F ` X ) ) >. /\ ( ( 1st ` ( `' F ` X ) ) e. J /\ ( 2nd ` ( `' F ` X ) ) e. NN0 ) ) )
11	10	simprd 114	. . . . 5 |- ( ( X e. NN /\ B e. Prime ) -> ( ( 1st ` ( `' F ` X ) ) e. J /\ ( 2nd ` ( `' F ` X ) ) e. NN0 ) )
12	11	simpld 112	. . . 4 |- ( ( X e. NN /\ B e. Prime ) -> ( 1st ` ( `' F ` X ) ) e. J )
13		breq2 4135	. . . . . 6 |- ( z = ( 1st ` ( `' F ` X ) ) -> ( B || z <-> B || ( 1st ` ( `' F ` X ) ) ) )
14	13	notbid 677	. . . . 5 |- ( z = ( 1st ` ( `' F ` X ) ) -> ( -. B || z <-> -. B || ( 1st ` ( `' F ` X ) ) ) )
15	14, 2	elrab2 2985	. . . 4 |- ( ( 1st ` ( `' F ` X ) ) e. J <-> ( ( 1st ` ( `' F ` X ) ) e. NN /\ -. B || ( 1st ` ( `' F ` X ) ) ) )
16	12, 15	sylib 122	. . 3 |- ( ( X e. NN /\ B e. Prime ) -> ( ( 1st ` ( `' F ` X ) ) e. NN /\ -. B || ( 1st ` ( `' F ` X ) ) ) )
17	16	simpld 112	. 2 |- ( ( X e. NN /\ B e. Prime ) -> ( 1st ` ( `' F ` X ) ) e. NN )
18	11	simprd 114	. 2 |- ( ( X e. NN /\ B e. Prime ) -> ( 2nd ` ( `' F ` X ) ) e. NN0 )
19	16	simprd 114	. 2 |- ( ( X e. NN /\ B e. Prime ) -> -. B || ( 1st ` ( `' F ` X ) ) )
20	10	simpld 112	. . . . 5 |- ( ( X e. NN /\ B e. Prime ) -> ( `' F ` X ) = <. ( 1st ` ( `' F ` X ) ) , ( 2nd ` ( `' F ` X ) ) >. )
21	20	fveq2d 5702	. . . 4 |- ( ( X e. NN /\ B e. Prime ) -> ( F ` ( `' F ` X ) ) = ( F ` <. ( 1st ` ( `' F ` X ) ) , ( 2nd ` ( `' F ` X ) ) >. ) )
22		df-ov 6091	. . . 4 |- ( ( 1st ` ( `' F ` X ) ) F ( 2nd ` ( `' F ` X ) ) ) = ( F ` <. ( 1st ` ( `' F ` X ) ) , ( 2nd ` ( `' F ` X ) ) >. )
23	21, 22	eqtr4di 2289	. . 3 |- ( ( X e. NN /\ B e. Prime ) -> ( F ` ( `' F ` X ) ) = ( ( 1st ` ( `' F ` X ) ) F ( 2nd ` ( `' F ` X ) ) ) )
24		f1ocnvfv2 5987	. . . 4 |- ( ( F : ( J X. NN0 ) -1-1-onto-> NN /\ X e. NN ) -> ( F ` ( `' F ` X ) ) = X )
25	5, 6, 24	syl2an2 602	. . 3 |- ( ( X e. NN /\ B e. Prime ) -> ( F ` ( `' F ` X ) ) = X )
26		prmnn 12913	. . . . . 6 |- ( B e. Prime -> B e. NN )
27		nnexpcl 11010	. . . . . 6 |- ( ( B e. NN /\ ( 2nd ` ( `' F ` X ) ) e. NN0 ) -> ( B ^ ( 2nd ` ( `' F ` X ) ) ) e. NN )
28	26, 18, 27	syl2an2 602	. . . . 5 |- ( ( X e. NN /\ B e. Prime ) -> ( B ^ ( 2nd ` ( `' F ` X ) ) ) e. NN )
29	28, 17	nnmulcld 9362	. . . 4 |- ( ( X e. NN /\ B e. Prime ) -> ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) e. NN )
30		oveq2 6096	. . . . 5 |- ( x = ( 1st ` ( `' F ` X ) ) -> ( ( B ^ y ) x. x ) = ( ( B ^ y ) x. ( 1st ` ( `' F ` X ) ) ) )
31		oveq2 6096	. . . . . 6 |- ( y = ( 2nd ` ( `' F ` X ) ) -> ( B ^ y ) = ( B ^ ( 2nd ` ( `' F ` X ) ) ) )
32	31	oveq1d 6103	. . . . 5 |- ( y = ( 2nd ` ( `' F ` X ) ) -> ( ( B ^ y ) x. ( 1st ` ( `' F ` X ) ) ) = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) )
33	30, 32, 3	ovmpog 6226	. . . 4 |- ( ( ( 1st ` ( `' F ` X ) ) e. J /\ ( 2nd ` ( `' F ` X ) ) e. NN0 /\ ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) e. NN ) -> ( ( 1st ` ( `' F ` X ) ) F ( 2nd ` ( `' F ` X ) ) ) = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) )
34	12, 18, 29, 33	syl3anc 1278	. . 3 |- ( ( X e. NN /\ B e. Prime ) -> ( ( 1st ` ( `' F ` X ) ) F ( 2nd ` ( `' F ` X ) ) ) = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) )
35	23, 25, 34	3eqtr3d 2279	. 2 |- ( ( X e. NN /\ B e. Prime ) -> X = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) )
36		breq2 4135	. . . . 5 |- ( m = ( 1st ` ( `' F ` X ) ) -> ( B || m <-> B || ( 1st ` ( `' F ` X ) ) ) )
37	36	notbid 677	. . . 4 |- ( m = ( 1st ` ( `' F ` X ) ) -> ( -. B || m <-> -. B || ( 1st ` ( `' F ` X ) ) ) )
38		oveq2 6096	. . . . 5 |- ( m = ( 1st ` ( `' F ` X ) ) -> ( ( B ^ a ) x. m ) = ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) )
39	38	eqeq2d 2250	. . . 4 |- ( m = ( 1st ` ( `' F ` X ) ) -> ( X = ( ( B ^ a ) x. m ) <-> X = ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) ) )
40	37, 39	anbi12d 477	. . 3 |- ( m = ( 1st ` ( `' F ` X ) ) -> ( ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) <-> ( -. B || ( 1st ` ( `' F ` X ) ) /\ X = ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) ) ) )
41		oveq2 6096	. . . . . 6 |- ( a = ( 2nd ` ( `' F ` X ) ) -> ( B ^ a ) = ( B ^ ( 2nd ` ( `' F ` X ) ) ) )
42	41	oveq1d 6103	. . . . 5 |- ( a = ( 2nd ` ( `' F ` X ) ) -> ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) )
43	42	eqeq2d 2250	. . . 4 |- ( a = ( 2nd ` ( `' F ` X ) ) -> ( X = ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) <-> X = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) ) )
44	43	anbi2d 468	. . 3 |- ( a = ( 2nd ` ( `' F ` X ) ) -> ( ( -. B || ( 1st ` ( `' F ` X ) ) /\ X = ( ( B ^ a ) x. ( 1st ` ( `' F ` X ) ) ) ) <-> ( -. B || ( 1st ` ( `' F ` X ) ) /\ X = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) ) ) )
45	40, 44	rspc2ev 2945	. 2 |- ( ( ( 1st ` ( `' F ` X ) ) e. NN /\ ( 2nd ` ( `' F ` X ) ) e. NN0 /\ ( -. B || ( 1st ` ( `' F ` X ) ) /\ X = ( ( B ^ ( 2nd ` ( `' F ` X ) ) ) x. ( 1st ` ( `' F ` X ) ) ) ) ) -> E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) )
46	17, 18, 19, 35, 45	syl112anc 1282	1 |- ( ( X e. NN /\ B e. Prime ) -> E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   X. cxp 4774   `'ccnv 4775   -1-1-onto->wf1o 5379   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   1stc1st 6376   2ndc2nd 6377   x. cmul 8190   NNcn 9313   2c2 9364   NN0cn0 9574   ZZ>=cuz 9937   ^cexp 10996   || cdvds 12579   Primecprime 12910

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-2o 6692  df-er 6811  df-en 7027  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-dvds 12580  df-prm 12911

This theorem is used by:  zprmlogbap  16218