Theorem logbgcd1irraplemap 15101

Description: Lemma for logbgcd1irrap 15102. 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 15100	. . . 4 |- ( ph -> ( X ^ N ) # ( B ^ M ) )
7		eluz2nn 9631	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	2, 7	syl 14	. . . . . . 7 |- ( ph -> B e. NN )
9	8	nnrpd 9760	. . . . . 6 |- ( ph -> B e. RR+ )
10		1red 8034	. . . . . . 7 |- ( ph -> 1 e. RR )
11	8	nnred 8995	. . . . . . 7 |- ( ph -> B e. RR )
12		eluz2gt1 9667	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
13	2, 12	syl 14	. . . . . . 7 |- ( ph -> 1 < B )
14	10, 11, 13	gtapd 8656	. . . . . 6 |- ( ph -> B # 1 )
15		eluz2nn 9631	. . . . . . . 8 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> X e. NN )
17	16	nnrpd 9760	. . . . . 6 |- ( ph -> X e. RR+ )
18		rpcxplogb 15096	. . . . . 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 5933	. . . 4 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) = ( X ^ N ) )
21		znq 9689	. . . . . . . 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 9690	. . . . . . 7 |- ( ( M / N ) e. QQ -> ( M / N ) e. RR )
24	22, 23	syl 14	. . . . . 6 |- ( ph -> ( M / N ) e. RR )
25	5	nncnd 8996	. . . . . 6 |- ( ph -> N e. CC )
26	9, 24, 25	cxpmuld 15070	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( ( B ^c ( M / N ) ) ^c N ) )
27	4	zcnd 9440	. . . . . . . 8 |- ( ph -> M e. CC )
28	5	nnap0d 9028	. . . . . . . 8 |- ( ph -> N # 0 )
29	27, 25, 28	divcanap1d 8810	. . . . . . 7 |- ( ph -> ( ( M / N ) x. N ) = M )
30	29	oveq2d 5934	. . . . . 6 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^c M ) )
31		cxpexpnn 15031	. . . . . . 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 2226	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^ M ) )
34	9, 24	rpcxpcld 15066	. . . . . 6 |- ( ph -> ( B ^c ( M / N ) ) e. RR+ )
35	5	nnzd 9438	. . . . . 6 |- ( ph -> N e. ZZ )
36		cxpexprp 15030	. . . . . 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 2235	. . . 4 |- ( ph -> ( ( B ^c ( M / N ) ) ^ N ) = ( B ^ M ) )
39	6, 20, 38	3brtr4d 4061	. . 3 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) )
40		relogbzcl 15084	. . . . . . 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 8048	. . . . 5 |- ( ph -> ( B logb X ) e. CC )
43	9, 42	rpcncxpcld 15061	. . . 4 |- ( ph -> ( B ^c ( B logb X ) ) e. CC )
44		qcn 9699	. . . . . 6 |- ( ( M / N ) e. QQ -> ( M / N ) e. CC )
45	22, 44	syl 14	. . . . 5 |- ( ph -> ( M / N ) e. CC )
46	9, 45	rpcncxpcld 15061	. . . 4 |- ( ph -> ( B ^c ( M / N ) ) e. CC )
47		apexp1 10789	. . . 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 15072	. . 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 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   1c1 7873   x. cmul 7877   < clt 8054   # cap 8600   / cdiv 8691   NNcn 8982   2c2 9033   ZZcz 9317   ZZ>=cuz 9592   QQcq 9684   RR+crp 9719   ^cexp 10609   gcd cgcd 12079   ^c ccxp 14992   log_b clogb 15075

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992  ax-pre-suploc 7993  ax-addf 7994  ax-mulf 7995

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-disj 4007  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-2o 6470  df-oadd 6473  df-er 6587  df-map 6704  df-pm 6705  df-en 6795  df-dom 6796  df-fin 6797  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-xneg 9838  df-xadd 9839  df-ioo 9958  df-ico 9960  df-icc 9961  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-bc 10819  df-ihash 10847  df-shft 10959  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497  df-ef 11791  df-e 11792  df-dvds 11931  df-gcd 12080  df-prm 12246  df-rest 12852  df-topgen 12871  df-psmet 14039  df-xmet 14040  df-met 14041  df-bl 14042  df-mopn 14043  df-top 14166  df-topon 14179  df-bases 14211  df-ntr 14264  df-cn 14356  df-cnp 14357  df-tx 14421  df-cncf 14726  df-limced 14810  df-dvap 14811  df-relog 14993  df-rpcxp 14994  df-logb 15076

This theorem is referenced by:  logbgcd1irrap  15102