Theorem zprmlogbap 16218

Description: The logarithm of a natural number to a prime base is either rational or irrational.

The proof decomposes X into m e. NN and a e. NN0 such that X = ( ( B ^ a ) x. m ) (using nnmaxpw 12978). If m = 1 the logarithm is a, which is rational. If 1 < m then we can apply logbgcd1irrap 16206 to show that the logarithm is irrational. (Contributed by Jim Kingdon and Taylor Barrella, 20-Aug-2026.)

Assertion

Ref	Expression
zprmlogbap	|- ( ( X e. NN /\ B e. Prime ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )

Distinct variable groups:   B, q   X, q

Proof of Theorem zprmlogbap

Dummy variables a m t u v x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4135	. . . . 5 |- ( t = z -> ( B || t <-> B || z ) )
2	1	notbid 677	. . . 4 |- ( t = z -> ( -. B || t <-> -. B || z ) )
3	2	cbvrabv 2820	. . 3 |- { t e. NN | -. B || t } = { z e. NN | -. B || z }
4		oveq2 6096	. . . 4 |- ( u = x -> ( ( B ^ v ) x. u ) = ( ( B ^ v ) x. x ) )
5		oveq2 6096	. . . . 5 |- ( v = y -> ( B ^ v ) = ( B ^ y ) )
6	5	oveq1d 6103	. . . 4 |- ( v = y -> ( ( B ^ v ) x. x ) = ( ( B ^ y ) x. x ) )
7	4, 6	cbvmpov 6171	. . 3 |- ( u e. { t e. NN | -. B || t } , v e. NN0 |-> ( ( B ^ v ) x. u ) ) = ( x e. { t e. NN | -. B || t } , y e. NN0 |-> ( ( B ^ y ) x. x ) )
8	3, 7	zprmlogbaplem3 16217	. 2 |- ( ( X e. NN /\ B e. Prime ) -> E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) )
9		simpllr 540	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> B e. Prime )
10		simplrl 541	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> m e. NN )
11		simprl 535	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> -. B || m )
12		simplrr 542	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> a e. NN0 )
13		eqid 2238	. . . . . 6 |- ( ( B ^ a ) x. m ) = ( ( B ^ a ) x. m )
14	9, 10, 11, 12, 13	zprmlogbaplem2 16216	. . . . 5 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( B logb ( ( B ^ a ) x. m ) ) e. QQ \/ ( ( B logb ( ( B ^ a ) x. m ) ) e. RR /\ A. q e. QQ ( B logb ( ( B ^ a ) x. m ) ) # q ) ) )
15		simprr 537	. . . . . . . 8 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> X = ( ( B ^ a ) x. m ) )
16	15	oveq2d 6104	. . . . . . 7 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( B logb X ) = ( B logb ( ( B ^ a ) x. m ) ) )
17	16	eleq1d 2307	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( B logb X ) e. QQ <-> ( B logb ( ( B ^ a ) x. m ) ) e. QQ ) )
18	16	eleq1d 2307	. . . . . . 7 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( B logb X ) e. RR <-> ( B logb ( ( B ^ a ) x. m ) ) e. RR ) )
19	16	breq1d 4141	. . . . . . . 8 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( B logb X ) # q <-> ( B logb ( ( B ^ a ) x. m ) ) # q ) )
20	19	ralbidv 2550	. . . . . . 7 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( A. q e. QQ ( B logb X ) # q <-> A. q e. QQ ( B logb ( ( B ^ a ) x. m ) ) # q ) )
21	18, 20	anbi12d 477	. . . . . 6 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) <-> ( ( B logb ( ( B ^ a ) x. m ) ) e. RR /\ A. q e. QQ ( B logb ( ( B ^ a ) x. m ) ) # q ) ) )
22	17, 21	orbi12d 805	. . . . 5 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) <-> ( ( B logb ( ( B ^ a ) x. m ) ) e. QQ \/ ( ( B logb ( ( B ^ a ) x. m ) ) e. RR /\ A. q e. QQ ( B logb ( ( B ^ a ) x. m ) ) # q ) ) ) )
23	14, 22	mpbird 167	. . . 4 |- ( ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) /\ ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )
24	23	ex 115	. . 3 |- ( ( ( X e. NN /\ B e. Prime ) /\ ( m e. NN /\ a e. NN0 ) ) -> ( ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) ) )
25	24	rexlimdvva 2676	. 2 |- ( ( X e. NN /\ B e. Prime ) -> ( E. m e. NN E. a e. NN0 ( -. B || m /\ X = ( ( B ^ a ) x. m ) ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) ) )
26	8, 25	mpd 13	1 |- ( ( X e. NN /\ B e. Prime ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {crab 2532   class class class wbr 4131  (class class class)co 6088   e. cmpo 6090   RRcr 8184   x. cmul 8190   # cap 8917   NNcn 9313   NN0cn0 9574   QQcq 10035   ^cexp 10996   || cdvds 12579   Primecprime 12910   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:  prmefexple  16308  bposlem1  16311