Theorem prmefexple 16308

Description: Convert a bound on a power of a prime to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Jim Kingdon, 21-Aug-2026.)

Assertion

Ref	Expression
prmefexple	|- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A ^ N ) <_ B <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )

Proof of Theorem prmefexple

Dummy variable q is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1028	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> A e. Prime )
2		prmnn 12913	. . . . . 6 |- ( A e. Prime -> A e. NN )
3	1, 2	syl 14	. . . . 5 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> A e. NN )
4	3	nnrpd 10112	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> A e. RR+ )
5		simp2 1029	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> N e. ZZ )
6		reexplog 16101	. . . 4 |- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) )
7	4, 5, 6	syl2anc 415	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) )
8		simp3 1030	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> B e. NN )
9	8	nnrpd 10112	. . . . 5 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> B e. RR+ )
10	9	reeflogd 16113	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( exp ` ( log ` B ) ) = B )
11	10	eqcomd 2244	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> B = ( exp ` ( log ` B ) ) )
12	7, 11	breq12d 4144	. 2 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A ^ N ) <_ B <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) )
13	5	zred 9779	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> N e. RR )
14	3	nnred 9326	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> A e. RR )
15		prmuz2 12935	. . . . . . . 8 |- ( A e. Prime -> A e. ( ZZ>= ` 2 ) )
16	1, 15	syl 14	. . . . . . 7 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> A e. ( ZZ>= ` 2 ) )
17		eluz2gt1 10018	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> 1 < A )
18	16, 17	syl 14	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> 1 < A )
19	14, 18	rplogcld 16118	. . . . 5 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( log ` A ) e. RR+ )
20	19	rpred 10114	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( log ` A ) e. RR )
21	13, 20	remulcld 8362	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( N x. ( log ` A ) ) e. RR )
22	9	relogcld 16112	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( log ` B ) e. RR )
23		efle 16002	. . 3 |- ( ( ( N x. ( log ` A ) ) e. RR /\ ( log ` B ) e. RR ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) )
24	21, 22, 23	syl2anc 415	. 2 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> ( exp ` ( N x. ( log ` A ) ) ) <_ ( exp ` ( log ` B ) ) ) )
25	13, 22, 19	lemuldivd 10164	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> N <_ ( ( log ` B ) / ( log ` A ) ) ) )
26		zprmlogbap 16218	. . . . . 6 |- ( ( B e. NN /\ A e. Prime ) -> ( ( A logb B ) e. QQ \/ ( ( A logb B ) e. RR /\ A. q e. QQ ( A logb B ) # q ) ) )
27	8, 1, 26	syl2anc 415	. . . . 5 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A logb B ) e. QQ \/ ( ( A logb B ) e. RR /\ A. q e. QQ ( A logb B ) # q ) ) )
28		relogbval 16187	. . . . . . . 8 |- ( ( A e. ( ZZ>= ` 2 ) /\ B e. RR+ ) -> ( A logb B ) = ( ( log ` B ) / ( log ` A ) ) )
29	16, 9, 28	syl2anc 415	. . . . . . 7 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( A logb B ) = ( ( log ` B ) / ( log ` A ) ) )
30	29	eleq1d 2307	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A logb B ) e. QQ <-> ( ( log ` B ) / ( log ` A ) ) e. QQ ) )
31	29	eleq1d 2307	. . . . . . 7 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A logb B ) e. RR <-> ( ( log ` B ) / ( log ` A ) ) e. RR ) )
32	29	breq1d 4141	. . . . . . . 8 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A logb B ) # q <-> ( ( log ` B ) / ( log ` A ) ) # q ) )
33	32	ralbidv 2550	. . . . . . 7 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( A. q e. QQ ( A logb B ) # q <-> A. q e. QQ ( ( log ` B ) / ( log ` A ) ) # q ) )
34	31, 33	anbi12d 477	. . . . . 6 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( ( A logb B ) e. RR /\ A. q e. QQ ( A logb B ) # q ) <-> ( ( ( log ` B ) / ( log ` A ) ) e. RR /\ A. q e. QQ ( ( log ` B ) / ( log ` A ) ) # q ) ) )
35	30, 34	orbi12d 805	. . . . 5 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( ( A logb B ) e. QQ \/ ( ( A logb B ) e. RR /\ A. q e. QQ ( A logb B ) # q ) ) <-> ( ( ( log ` B ) / ( log ` A ) ) e. QQ \/ ( ( ( log ` B ) / ( log ` A ) ) e. RR /\ A. q e. QQ ( ( log ` B ) / ( log ` A ) ) # q ) ) ) )
36	27, 35	mpbid 147	. . . 4 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( ( log ` B ) / ( log ` A ) ) e. QQ \/ ( ( ( log ` B ) / ( log ` A ) ) e. RR /\ A. q e. QQ ( ( log ` B ) / ( log ` A ) ) # q ) ) )
37		flapge 10737	. . . 4 |- ( ( ( ( ( log ` B ) / ( log ` A ) ) e. QQ \/ ( ( ( log ` B ) / ( log ` A ) ) e. RR /\ A. q e. QQ ( ( log ` B ) / ( log ` A ) ) # q ) ) /\ N e. ZZ ) -> ( N <_ ( ( log ` B ) / ( log ` A ) ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )
38	36, 5, 37	syl2anc 415	. . 3 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( N <_ ( ( log ` B ) / ( log ` A ) ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )
39	25, 38	bitrd 188	. 2 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( N x. ( log ` A ) ) <_ ( log ` B ) <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )
40	12, 24, 39	3bitr2d 216	1 |- ( ( A e. Prime /\ N e. ZZ /\ B e. NN ) -> ( ( A ^ N ) <_ B <-> N <_ ( |_ ` ( ( log ` B ) / ( log ` A ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   1c1 8186   x. cmul 8190   < clt 8366   <_ cle 8367   # cap 8917   / cdiv 9010   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   QQcq 10035   RR+crp 10071   |_cfl 10720   ^cexp 10996   expce 12434   Primecprime 12910   logclog 16085   log_b clogb 16179

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  ax-pre-suploc 8306  ax-addf 8307  ax-mulf 8308

This proof depends on definitions:  df-bi 117  df-stab 843  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-disj 4107  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-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7027  df-dom 7028  df-fin 7029  df-sup 7330  df-inf 7331  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-xneg 10191  df-xadd 10192  df-ioo 10311  df-ico 10313  df-icc 10314  df-fz 10429  df-fzo 10567  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-fac 11186  df-bc 11208  df-ihash 11237  df-shft 11602  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145  df-ef 12440  df-e 12441  df-dvds 12580  df-gcd 12756  df-prm 12911  df-rest 13654  df-topgen 13673  df-psmet 14989  df-xmet 14990  df-met 14991  df-bl 14992  df-mopn 14993  df-top 15224  df-topon 15237  df-bases 15269  df-ntr 15322  df-cn 15414  df-cnp 15415  df-tx 15479  df-cncf 15797  df-limced 15882  df-dvap 15883  df-relog 16087  df-rpcxp 16088  df-logb 16180

This theorem is used by:  bposlem1  16311