Theorem zprmlogbaplem2 16135

Description: Lemma for zprmlogbap 16137. The logarithm is either rational or irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)

Hypotheses

Ref	Expression
zprmlogbaplem1.b	|- ( ph -> B e. Prime )
zprmlogbaplem1.m	|- ( ph -> M e. NN )
zprmlogbaplem1.j	|- ( ph -> -. B || M )
zprmlogbaplem1.a	|- ( ph -> A e. NN0 )
zprmlogbaplem2.x	|- X = ( ( B ^ A ) x. M )

Assertion

Ref	Expression
zprmlogbaplem2	|- ( ph -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )

Distinct variable groups:   A, q   B, q   M, q   X, q   ph, q

Proof of Theorem zprmlogbaplem2

Step	Hyp	Ref	Expression
1		zprmlogbaplem1.m	. . 3 |- ( ph -> M e. NN )
2		elnn1uz2 10016	. . 3 |- ( M e. NN <-> ( M = 1 \/ M e. ( ZZ>= ` 2 ) ) )
3	1, 2	sylib 122	. 2 |- ( ph -> ( M = 1 \/ M e. ( ZZ>= ` 2 ) ) )
4		zprmlogbaplem2.x	. . . . . . . . 9 |- X = ( ( B ^ A ) x. M )
5	4	oveq2i 6096	. . . . . . . 8 |- ( B logb X ) = ( B logb ( ( B ^ A ) x. M ) )
6		zprmlogbaplem1.b	. . . . . . . . 9 |- ( ph -> B e. Prime )
7		zprmlogbaplem1.j	. . . . . . . . 9 |- ( ph -> -. B || M )
8		zprmlogbaplem1.a	. . . . . . . . 9 |- ( ph -> A e. NN0 )
9	6, 1, 7, 8	zprmlogbaplem1 16134	. . . . . . . 8 |- ( ph -> ( B logb ( ( B ^ A ) x. M ) ) = ( A + ( B logb M ) ) )
10	5, 9	eqtrid 2283	. . . . . . 7 |- ( ph -> ( B logb X ) = ( A + ( B logb M ) ) )
11	10	adantr 276	. . . . . 6 |- ( ( ph /\ M = 1 ) -> ( B logb X ) = ( A + ( B logb M ) ) )
12		oveq2 6093	. . . . . . . 8 |- ( M = 1 -> ( B logb M ) = ( B logb 1 ) )
13		prmnn 12904	. . . . . . . . . . 11 |- ( B e. Prime -> B e. NN )
14	6, 13	syl 14	. . . . . . . . . 10 |- ( ph -> B e. NN )
15	14	nnrpd 10105	. . . . . . . . 9 |- ( ph -> B e. RR+ )
16		1red 8341	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
17	14	nnred 9319	. . . . . . . . . 10 |- ( ph -> B e. RR )
18		prmgt1 12927	. . . . . . . . . . 11 |- ( B e. Prime -> 1 < B )
19	6, 18	syl 14	. . . . . . . . . 10 |- ( ph -> 1 < B )
20	16, 17, 19	gtapd 8967	. . . . . . . . 9 |- ( ph -> B # 1 )
21		rplogb1 16103	. . . . . . . . 9 |- ( ( B e. RR+ /\ B # 1 ) -> ( B logb 1 ) = 0 )
22	15, 20, 21	syl2anc 415	. . . . . . . 8 |- ( ph -> ( B logb 1 ) = 0 )
23	12, 22	sylan9eqr 2293	. . . . . . 7 |- ( ( ph /\ M = 1 ) -> ( B logb M ) = 0 )
24	23	oveq2d 6101	. . . . . 6 |- ( ( ph /\ M = 1 ) -> ( A + ( B logb M ) ) = ( A + 0 ) )
25	8	nn0cnd 9626	. . . . . . . 8 |- ( ph -> A e. CC )
26	25	adantr 276	. . . . . . 7 |- ( ( ph /\ M = 1 ) -> A e. CC )
27	26	addridd 8476	. . . . . 6 |- ( ( ph /\ M = 1 ) -> ( A + 0 ) = A )
28	11, 24, 27	3eqtrd 2275	. . . . 5 |- ( ( ph /\ M = 1 ) -> ( B logb X ) = A )
29	8	nn0zd 9770	. . . . . . 7 |- ( ph -> A e. ZZ )
30		zq 10035	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
31	29, 30	syl 14	. . . . . 6 |- ( ph -> A e. QQ )
32	31	adantr 276	. . . . 5 |- ( ( ph /\ M = 1 ) -> A e. QQ )
33	28, 32	eqeltrd 2315	. . . 4 |- ( ( ph /\ M = 1 ) -> ( B logb X ) e. QQ )
34	33	ex 115	. . 3 |- ( ph -> ( M = 1 -> ( B logb X ) e. QQ ) )
35		prmuz2 12926	. . . . . . . . . . 11 |- ( B e. Prime -> B e. ( ZZ>= ` 2 ) )
36	6, 35	syl 14	. . . . . . . . . 10 |- ( ph -> B e. ( ZZ>= ` 2 ) )
37	1	nnrpd 10105	. . . . . . . . . 10 |- ( ph -> M e. RR+ )
38		relogbzcl 16107	. . . . . . . . . 10 |- ( ( B e. ( ZZ>= ` 2 ) /\ M e. RR+ ) -> ( B logb M ) e. RR )
39	36, 37, 38	syl2anc 415	. . . . . . . . 9 |- ( ph -> ( B logb M ) e. RR )
40	39	recnd 8354	. . . . . . . 8 |- ( ph -> ( B logb M ) e. CC )
41	25, 40, 10	comraddd 8484	. . . . . . 7 |- ( ph -> ( B logb X ) = ( ( B logb M ) + A ) )
42	41	adantr 276	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( B logb X ) = ( ( B logb M ) + A ) )
43	39	adantr 276	. . . . . . . 8 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( B logb M ) e. RR )
44		simpr 110	. . . . . . . . 9 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> M e. ( ZZ>= ` 2 ) )
45	36	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> B e. ( ZZ>= ` 2 ) )
46	1	nnzd 9771	. . . . . . . . . . . 12 |- ( ph -> M e. ZZ )
47	14	nnzd 9771	. . . . . . . . . . . 12 |- ( ph -> B e. ZZ )
48	46, 47	gcdcomd 12767	. . . . . . . . . . 11 |- ( ph -> ( M gcd B ) = ( B gcd M ) )
49		coprm 12939	. . . . . . . . . . . . 13 |- ( ( B e. Prime /\ M e. ZZ ) -> ( -. B || M <-> ( B gcd M ) = 1 ) )
50	6, 46, 49	syl2anc 415	. . . . . . . . . . . 12 |- ( ph -> ( -. B || M <-> ( B gcd M ) = 1 ) )
51	7, 50	mpbid 147	. . . . . . . . . . 11 |- ( ph -> ( B gcd M ) = 1 )
52	48, 51	eqtrd 2271	. . . . . . . . . 10 |- ( ph -> ( M gcd B ) = 1 )
53	52	adantr 276	. . . . . . . . 9 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( M gcd B ) = 1 )
54		logbgcd1irrap 16125	. . . . . . . . . . 11 |- ( ( ( M e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( M gcd B ) = 1 /\ q e. QQ ) ) -> ( B logb M ) # q )
55	54	anassrs 404	. . . . . . . . . 10 |- ( ( ( ( M e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( M gcd B ) = 1 ) /\ q e. QQ ) -> ( B logb M ) # q )
56	55	ralrimiva 2623	. . . . . . . . 9 |- ( ( ( M e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( M gcd B ) = 1 ) -> A. q e. QQ ( B logb M ) # q )
57	44, 45, 53, 56	syl21anc 1277	. . . . . . . 8 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> A. q e. QQ ( B logb M ) # q )
58	31	adantr 276	. . . . . . . 8 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> A e. QQ )
59		irraddap 10056	. . . . . . . 8 |- ( ( ( ( B logb M ) e. RR /\ A. q e. QQ ( B logb M ) # q ) /\ A e. QQ ) -> ( ( ( B logb M ) + A ) e. RR /\ A. q e. QQ ( ( B logb M ) + A ) # q ) )
60	43, 57, 58, 59	syl21anc 1277	. . . . . . 7 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( ( ( B logb M ) + A ) e. RR /\ A. q e. QQ ( ( B logb M ) + A ) # q ) )
61	60	simpld 112	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( ( B logb M ) + A ) e. RR )
62	42, 61	eqeltrd 2315	. . . . 5 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( B logb X ) e. RR )
63	60	simprd 114	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> A. q e. QQ ( ( B logb M ) + A ) # q )
64	41	breq1d 4140	. . . . . . . 8 |- ( ph -> ( ( B logb X ) # q <-> ( ( B logb M ) + A ) # q ) )
65	64	ralbidv 2550	. . . . . . 7 |- ( ph -> ( A. q e. QQ ( B logb X ) # q <-> A. q e. QQ ( ( B logb M ) + A ) # q ) )
66	65	adantr 276	. . . . . 6 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( A. q e. QQ ( B logb X ) # q <-> A. q e. QQ ( ( B logb M ) + A ) # q ) )
67	63, 66	mpbird 167	. . . . 5 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> A. q e. QQ ( B logb X ) # q )
68	62, 67	jca 306	. . . 4 |- ( ( ph /\ M e. ( ZZ>= ` 2 ) ) -> ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) )
69	68	ex 115	. . 3 |- ( ph -> ( M e. ( ZZ>= ` 2 ) -> ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) )
70	34, 69	orim12d 798	. 2 |- ( ph -> ( ( M = 1 \/ M e. ( ZZ>= ` 2 ) ) -> ( ( B logb X ) e. QQ \/ ( ( B logb X ) e. RR /\ A. q e. QQ ( B logb X ) # q ) ) ) )
71	3, 70	mpd 13	1 |- ( ph -> ( ( 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   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   # cap 8911   NNcn 9306   2c2 9357   NN0cn0 9567   ZZcz 9648   ZZ>=cuz 9930   QQcq 10028   RR+crp 10064   ^cexp 10988   || cdvds 12570   gcd cgcd 12746   Primecprime 12901   log_b clogb 16098

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  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

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 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-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  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-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-xneg 10184  df-xadd 10185  df-ioo 10304  df-ico 10306  df-icc 10307  df-fz 10422  df-fzo 10560  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-fac 11178  df-bc 11200  df-ihash 11229  df-shft 11594  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-ef 12431  df-e 12432  df-dvds 12571  df-gcd 12747  df-prm 12902  df-rest 13644  df-topgen 13663  df-psmet 14929  df-xmet 14930  df-met 14931  df-bl 14932  df-mopn 14933  df-top 15148  df-topon 15161  df-bases 15193  df-ntr 15246  df-cn 15338  df-cnp 15339  df-tx 15403  df-cncf 15721  df-limced 15806  df-dvap 15807  df-relog 16009  df-rpcxp 16010  df-logb 16099

This theorem is used by:  zprmlogbap  16137