Theorem logbgcd1irraplemap 15359

Description: Lemma for logbgcd1irrap 15360. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)

Hypotheses

Ref	Expression
logbgcd1irraplem.x	|- ( ph -> X e. ( ZZ>= ` 2 ) )
logbgcd1irraplem.b	|- ( ph -> B e. ( ZZ>= ` 2 ) )
logbgcd1irraplem.rp	|- ( ph -> ( X gcd B ) = 1 )
logbgcd1irraplem.m	|- ( ph -> M e. ZZ )
logbgcd1irraplem.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
logbgcd1irraplemap	|- ( ph -> ( B logb X ) # ( M / N ) )

Proof of Theorem logbgcd1irraplemap

Step	Hyp	Ref	Expression
1		logbgcd1irraplem.x	. . . . 5 |- ( ph -> X e. ( ZZ>= ` 2 ) )
2		logbgcd1irraplem.b	. . . . 5 |- ( ph -> B e. ( ZZ>= ` 2 ) )
3		logbgcd1irraplem.rp	. . . . 5 |- ( ph -> ( X gcd B ) = 1 )
4		logbgcd1irraplem.m	. . . . 5 |- ( ph -> M e. ZZ )
5		logbgcd1irraplem.n	. . . . 5 |- ( ph -> N e. NN )
6	1, 2, 3, 4, 5	logbgcd1irraplemexp 15358	. . . 4 |- ( ph -> ( X ^ N ) # ( B ^ M ) )
7		eluz2nn 9669	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	2, 7	syl 14	. . . . . . 7 |- ( ph -> B e. NN )
9	8	nnrpd 9798	. . . . . 6 |- ( ph -> B e. RR+ )
10		1red 8069	. . . . . . 7 |- ( ph -> 1 e. RR )
11	8	nnred 9031	. . . . . . 7 |- ( ph -> B e. RR )
12		eluz2gt1 9705	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
13	2, 12	syl 14	. . . . . . 7 |- ( ph -> 1 < B )
14	10, 11, 13	gtapd 8692	. . . . . 6 |- ( ph -> B # 1 )
15		eluz2nn 9669	. . . . . . . 8 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> X e. NN )
17	16	nnrpd 9798	. . . . . 6 |- ( ph -> X e. RR+ )
18		rpcxplogb 15354	. . . . . 6 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B ^c ( B logb X ) ) = X )
19	9, 14, 17, 18	syl3anc 1249	. . . . 5 |- ( ph -> ( B ^c ( B logb X ) ) = X )
20	19	oveq1d 5949	. . . 4 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) = ( X ^ N ) )
21		znq 9727	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
22	4, 5, 21	syl2anc 411	. . . . . . 7 |- ( ph -> ( M / N ) e. QQ )
23		qre 9728	. . . . . . 7 |- ( ( M / N ) e. QQ -> ( M / N ) e. RR )
24	22, 23	syl 14	. . . . . 6 |- ( ph -> ( M / N ) e. RR )
25	5	nncnd 9032	. . . . . 6 |- ( ph -> N e. CC )
26	9, 24, 25	cxpmuld 15327	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( ( B ^c ( M / N ) ) ^c N ) )
27	4	zcnd 9478	. . . . . . . 8 |- ( ph -> M e. CC )
28	5	nnap0d 9064	. . . . . . . 8 |- ( ph -> N # 0 )
29	27, 25, 28	divcanap1d 8846	. . . . . . 7 |- ( ph -> ( ( M / N ) x. N ) = M )
30	29	oveq2d 5950	. . . . . 6 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^c M ) )
31		cxpexpnn 15286	. . . . . . 7 |- ( ( B e. NN /\ M e. ZZ ) -> ( B ^c M ) = ( B ^ M ) )
32	8, 4, 31	syl2anc 411	. . . . . 6 |- ( ph -> ( B ^c M ) = ( B ^ M ) )
33	30, 32	eqtrd 2237	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^ M ) )
34	9, 24	rpcxpcld 15323	. . . . . 6 |- ( ph -> ( B ^c ( M / N ) ) e. RR+ )
35	5	nnzd 9476	. . . . . 6 |- ( ph -> N e. ZZ )
36		cxpexprp 15285	. . . . . 6 |- ( ( ( B ^c ( M / N ) ) e. RR+ /\ N e. ZZ ) -> ( ( B ^c ( M / N ) ) ^c N ) = ( ( B ^c ( M / N ) ) ^ N ) )
37	34, 35, 36	syl2anc 411	. . . . 5 |- ( ph -> ( ( B ^c ( M / N ) ) ^c N ) = ( ( B ^c ( M / N ) ) ^ N ) )
38	26, 33, 37	3eqtr3rd 2246	. . . 4 |- ( ph -> ( ( B ^c ( M / N ) ) ^ N ) = ( B ^ M ) )
39	6, 20, 38	3brtr4d 4075	. . 3 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) )
40		relogbzcl 15342	. . . . . . 7 |- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) e. RR )
41	2, 17, 40	syl2anc 411	. . . . . 6 |- ( ph -> ( B logb X ) e. RR )
42	41	recnd 8083	. . . . 5 |- ( ph -> ( B logb X ) e. CC )
43	9, 42	rpcncxpcld 15317	. . . 4 |- ( ph -> ( B ^c ( B logb X ) ) e. CC )
44		qcn 9737	. . . . . 6 |- ( ( M / N ) e. QQ -> ( M / N ) e. CC )
45	22, 44	syl 14	. . . . 5 |- ( ph -> ( M / N ) e. CC )
46	9, 45	rpcncxpcld 15317	. . . 4 |- ( ph -> ( B ^c ( M / N ) ) e. CC )
47		apexp1 10844	. . . 4 |- ( ( ( B ^c ( B logb X ) ) e. CC /\ ( B ^c ( M / N ) ) e. CC /\ N e. NN ) -> ( ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) -> ( B ^c ( B logb X ) ) # ( B ^c ( M / N ) ) ) )
48	43, 46, 5, 47	syl3anc 1249	. . 3 |- ( ph -> ( ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) -> ( B ^c ( B logb X ) ) # ( B ^c ( M / N ) ) ) )
49	39, 48	mpd 13	. 2 |- ( ph -> ( B ^c ( B logb X ) ) # ( B ^c ( M / N ) ) )
50		apcxp2 15329	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( ( B logb X ) e. RR /\ ( M / N ) e. RR ) ) -> ( ( B logb X ) # ( M / N ) <-> ( B ^c ( B logb X ) ) # ( B ^c ( M / N ) ) ) )
51	9, 14, 41, 24, 50	syl22anc 1250	. 2 |- ( ph -> ( ( B logb X ) # ( M / N ) <-> ( B ^c ( B logb X ) ) # ( B ^c ( M / N ) ) ) )
52	49, 51	mpbird 167	1 |- ( ph -> ( B logb X ) # ( M / N ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1372   e. wcel 2175   class class class wbr 4043   ` cfv 5268  (class class class)co 5934   CCcc 7905   RRcr 7906   1c1 7908   x. cmul 7912   < clt 8089   # cap 8636   / cdiv 8727   NNcn 9018   2c2 9069   ZZcz 9354   ZZ>=cuz 9630   QQcq 9722   RR+crp 9757   ^cexp 10664   gcd cgcd 12193   ^c ccxp 15247   log_b clogb 15333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634  ax-cnex 7998  ax-resscn 7999  ax-1cn 8000  ax-1re 8001  ax-icn 8002  ax-addcl 8003  ax-addrcl 8004  ax-mulcl 8005  ax-mulrcl 8006  ax-addcom 8007  ax-mulcom 8008  ax-addass 8009  ax-mulass 8010  ax-distr 8011  ax-i2m1 8012  ax-0lt1 8013  ax-1rid 8014  ax-0id 8015  ax-rnegex 8016  ax-precex 8017  ax-cnre 8018  ax-pre-ltirr 8019  ax-pre-ltwlin 8020  ax-pre-lttrn 8021  ax-pre-apti 8022  ax-pre-ltadd 8023  ax-pre-mulgt0 8024  ax-pre-mulext 8025  ax-arch 8026  ax-caucvg 8027  ax-pre-suploc 8028  ax-addf 8029  ax-mulf 8030

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-disj 4021  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-ilim 4414  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-isom 5277  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-of 6148  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-frec 6467  df-1o 6492  df-2o 6493  df-oadd 6496  df-er 6610  df-map 6727  df-pm 6728  df-en 6818  df-dom 6819  df-fin 6820  df-sup 7068  df-inf 7069  df-pnf 8091  df-mnf 8092  df-xr 8093  df-ltxr 8094  df-le 8095  df-sub 8227  df-neg 8228  df-reap 8630  df-ap 8637  df-div 8728  df-inn 9019  df-2 9077  df-3 9078  df-4 9079  df-n0 9278  df-z 9355  df-uz 9631  df-q 9723  df-rp 9758  df-xneg 9876  df-xadd 9877  df-ioo 9996  df-ico 9998  df-icc 9999  df-fz 10113  df-fzo 10247  df-fl 10394  df-mod 10449  df-seqfrec 10574  df-exp 10665  df-fac 10852  df-bc 10874  df-ihash 10902  df-shft 11045  df-cj 11072  df-re 11073  df-im 11074  df-rsqrt 11228  df-abs 11229  df-clim 11509  df-sumdc 11584  df-ef 11878  df-e 11879  df-dvds 12018  df-gcd 12194  df-prm 12349  df-rest 12991  df-topgen 13010  df-psmet 14223  df-xmet 14224  df-met 14225  df-bl 14226  df-mopn 14227  df-top 14388  df-topon 14401  df-bases 14433  df-ntr 14486  df-cn 14578  df-cnp 14579  df-tx 14643  df-cncf 14961  df-limced 15046  df-dvap 15047  df-relog 15248  df-rpcxp 15249  df-logb 15334

This theorem is referenced by:  logbgcd1irrap  15360