Theorem logbgcd1irraplemap 16205

Description: Lemma for logbgcd1irrap 16206. 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 16204	. . . 4 |- ( ph -> ( X ^ N ) # ( B ^ M ) )
7		eluz2nn 9982	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	2, 7	syl 14	. . . . . . 7 |- ( ph -> B e. NN )
9	8	nnrpd 10112	. . . . . 6 |- ( ph -> B e. RR+ )
10		1red 8347	. . . . . . 7 |- ( ph -> 1 e. RR )
11	8	nnred 9326	. . . . . . 7 |- ( ph -> B e. RR )
12		eluz2gt1 10018	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
13	2, 12	syl 14	. . . . . . 7 |- ( ph -> 1 < B )
14	10, 11, 13	gtapd 8973	. . . . . 6 |- ( ph -> B # 1 )
15		eluz2nn 9982	. . . . . . . 8 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> X e. NN )
17	16	nnrpd 10112	. . . . . 6 |- ( ph -> X e. RR+ )
18		rpcxplogb 16200	. . . . . 6 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B ^c ( B logb X ) ) = X )
19	9, 14, 17, 18	syl3anc 1278	. . . . 5 |- ( ph -> ( B ^c ( B logb X ) ) = X )
20	19	oveq1d 6103	. . . 4 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) = ( X ^ N ) )
21		znq 10040	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. NN ) -> ( M / N ) e. QQ )
22	4, 5, 21	syl2anc 415	. . . . . . 7 |- ( ph -> ( M / N ) e. QQ )
23		qre 10041	. . . . . . 7 |- ( ( M / N ) e. QQ -> ( M / N ) e. RR )
24	22, 23	syl 14	. . . . . 6 |- ( ph -> ( M / N ) e. RR )
25	5	nncnd 9327	. . . . . 6 |- ( ph -> N e. CC )
26	9, 24, 25	cxpmuld 16172	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( ( B ^c ( M / N ) ) ^c N ) )
27	4	zcnd 9780	. . . . . . . 8 |- ( ph -> M e. CC )
28	5	nnap0d 9359	. . . . . . . 8 |- ( ph -> N # 0 )
29	27, 25, 28	divcanap1d 9129	. . . . . . 7 |- ( ph -> ( ( M / N ) x. N ) = M )
30	29	oveq2d 6104	. . . . . 6 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^c M ) )
31		cxpexpnn 16129	. . . . . . 7 |- ( ( B e. NN /\ M e. ZZ ) -> ( B ^c M ) = ( B ^ M ) )
32	8, 4, 31	syl2anc 415	. . . . . 6 |- ( ph -> ( B ^c M ) = ( B ^ M ) )
33	30, 32	eqtrd 2271	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^ M ) )
34	9, 24	rpcxpcld 16168	. . . . . 6 |- ( ph -> ( B ^c ( M / N ) ) e. RR+ )
35	5	nnzd 9778	. . . . . 6 |- ( ph -> N e. ZZ )
36		cxpexprp 16128	. . . . . 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 415	. . . . 5 |- ( ph -> ( ( B ^c ( M / N ) ) ^c N ) = ( ( B ^c ( M / N ) ) ^ N ) )
38	26, 33, 37	3eqtr3rd 2280	. . . 4 |- ( ph -> ( ( B ^c ( M / N ) ) ^ N ) = ( B ^ M ) )
39	6, 20, 38	3brtr4d 4163	. . 3 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) )
40		relogbzcl 16188	. . . . . . 7 |- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) e. RR )
41	2, 17, 40	syl2anc 415	. . . . . 6 |- ( ph -> ( B logb X ) e. RR )
42	41	recnd 8360	. . . . 5 |- ( ph -> ( B logb X ) e. CC )
43	9, 42	rpcncxpcld 16162	. . . 4 |- ( ph -> ( B ^c ( B logb X ) ) e. CC )
44		qcn 10050	. . . . . 6 |- ( ( M / N ) e. QQ -> ( M / N ) e. CC )
45	22, 44	syl 14	. . . . 5 |- ( ph -> ( M / N ) e. CC )
46	9, 45	rpcncxpcld 16162	. . . 4 |- ( ph -> ( B ^c ( M / N ) ) e. CC )
47		apexp1 11178	. . . 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 1278	. . 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 16174	. . 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 1279	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   1c1 8186   x. cmul 8190   < clt 8366   # cap 8917   / cdiv 9010   NNcn 9313   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   QQcq 10035   RR+crp 10071   ^cexp 10996   gcd cgcd 12755   ^c ccxp 16086   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:  logbgcd1irrap  16206