Theorem logbgcd1irraplemap 15313

Description: Lemma for logbgcd1irrap 15314. 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 15312	. . . 4 |- ( ph -> ( X ^ N ) # ( B ^ M ) )
7		eluz2nn 9659	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	2, 7	syl 14	. . . . . . 7 |- ( ph -> B e. NN )
9	8	nnrpd 9788	. . . . . 6 |- ( ph -> B e. RR+ )
10		1red 8060	. . . . . . 7 |- ( ph -> 1 e. RR )
11	8	nnred 9022	. . . . . . 7 |- ( ph -> B e. RR )
12		eluz2gt1 9695	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
13	2, 12	syl 14	. . . . . . 7 |- ( ph -> 1 < B )
14	10, 11, 13	gtapd 8683	. . . . . 6 |- ( ph -> B # 1 )
15		eluz2nn 9659	. . . . . . . 8 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> X e. NN )
17	16	nnrpd 9788	. . . . . 6 |- ( ph -> X e. RR+ )
18		rpcxplogb 15308	. . . . . 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 5940	. . . 4 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) = ( X ^ N ) )
21		znq 9717	. . . . . . . 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 9718	. . . . . . 7 |- ( ( M / N ) e. QQ -> ( M / N ) e. RR )
24	22, 23	syl 14	. . . . . 6 |- ( ph -> ( M / N ) e. RR )
25	5	nncnd 9023	. . . . . 6 |- ( ph -> N e. CC )
26	9, 24, 25	cxpmuld 15281	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( ( B ^c ( M / N ) ) ^c N ) )
27	4	zcnd 9468	. . . . . . . 8 |- ( ph -> M e. CC )
28	5	nnap0d 9055	. . . . . . . 8 |- ( ph -> N # 0 )
29	27, 25, 28	divcanap1d 8837	. . . . . . 7 |- ( ph -> ( ( M / N ) x. N ) = M )
30	29	oveq2d 5941	. . . . . 6 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^c M ) )
31		cxpexpnn 15240	. . . . . . 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 2229	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^ M ) )
34	9, 24	rpcxpcld 15277	. . . . . 6 |- ( ph -> ( B ^c ( M / N ) ) e. RR+ )
35	5	nnzd 9466	. . . . . 6 |- ( ph -> N e. ZZ )
36		cxpexprp 15239	. . . . . 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 2238	. . . 4 |- ( ph -> ( ( B ^c ( M / N ) ) ^ N ) = ( B ^ M ) )
39	6, 20, 38	3brtr4d 4066	. . 3 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) )
40		relogbzcl 15296	. . . . . . 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 8074	. . . . 5 |- ( ph -> ( B logb X ) e. CC )
43	9, 42	rpcncxpcld 15271	. . . 4 |- ( ph -> ( B ^c ( B logb X ) ) e. CC )
44		qcn 9727	. . . . . 6 |- ( ( M / N ) e. QQ -> ( M / N ) e. CC )
45	22, 44	syl 14	. . . . 5 |- ( ph -> ( M / N ) e. CC )
46	9, 45	rpcncxpcld 15271	. . . 4 |- ( ph -> ( B ^c ( M / N ) ) e. CC )
47		apexp1 10829	. . . 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 15283	. . 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 2167   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7896   RRcr 7897   1c1 7899   x. cmul 7903   < clt 8080   # cap 8627   / cdiv 8718   NNcn 9009   2c2 9060   ZZcz 9345   ZZ>=cuz 9620   QQcq 9712   RR+crp 9747   ^cexp 10649   gcd cgcd 12147   ^c ccxp 15201   log_b clogb 15287

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-mulrcl 7997  ax-addcom 7998  ax-mulcom 7999  ax-addass 8000  ax-mulass 8001  ax-distr 8002  ax-i2m1 8003  ax-0lt1 8004  ax-1rid 8005  ax-0id 8006  ax-rnegex 8007  ax-precex 8008  ax-cnre 8009  ax-pre-ltirr 8010  ax-pre-ltwlin 8011  ax-pre-lttrn 8012  ax-pre-apti 8013  ax-pre-ltadd 8014  ax-pre-mulgt0 8015  ax-pre-mulext 8016  ax-arch 8017  ax-caucvg 8018  ax-pre-suploc 8019  ax-addf 8020  ax-mulf 8021

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-2o 6484  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8082  df-mnf 8083  df-xr 8084  df-ltxr 8085  df-le 8086  df-sub 8218  df-neg 8219  df-reap 8621  df-ap 8628  df-div 8719  df-inn 9010  df-2 9068  df-3 9069  df-4 9070  df-n0 9269  df-z 9346  df-uz 9621  df-q 9713  df-rp 9748  df-xneg 9866  df-xadd 9867  df-ioo 9986  df-ico 9988  df-icc 9989  df-fz 10103  df-fzo 10237  df-fl 10379  df-mod 10434  df-seqfrec 10559  df-exp 10650  df-fac 10837  df-bc 10859  df-ihash 10887  df-shft 10999  df-cj 11026  df-re 11027  df-im 11028  df-rsqrt 11182  df-abs 11183  df-clim 11463  df-sumdc 11538  df-ef 11832  df-e 11833  df-dvds 11972  df-gcd 12148  df-prm 12303  df-rest 12945  df-topgen 12964  df-psmet 14177  df-xmet 14178  df-met 14179  df-bl 14180  df-mopn 14181  df-top 14342  df-topon 14355  df-bases 14387  df-ntr 14440  df-cn 14532  df-cnp 14533  df-tx 14597  df-cncf 14915  df-limced 15000  df-dvap 15001  df-relog 15202  df-rpcxp 15203  df-logb 15288

This theorem is referenced by:  logbgcd1irrap  15314