Theorem logbgcd1irraplemap 15658

Description: Lemma for logbgcd1irrap 15659. 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 15657	. . . 4 |- ( ph -> ( X ^ N ) # ( B ^ M ) )
7		eluz2nn 9773	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> B e. NN )
8	2, 7	syl 14	. . . . . . 7 |- ( ph -> B e. NN )
9	8	nnrpd 9902	. . . . . 6 |- ( ph -> B e. RR+ )
10		1red 8172	. . . . . . 7 |- ( ph -> 1 e. RR )
11	8	nnred 9134	. . . . . . 7 |- ( ph -> B e. RR )
12		eluz2gt1 9809	. . . . . . . 8 |- ( B e. ( ZZ>= ` 2 ) -> 1 < B )
13	2, 12	syl 14	. . . . . . 7 |- ( ph -> 1 < B )
14	10, 11, 13	gtapd 8795	. . . . . 6 |- ( ph -> B # 1 )
15		eluz2nn 9773	. . . . . . . 8 |- ( X e. ( ZZ>= ` 2 ) -> X e. NN )
16	1, 15	syl 14	. . . . . . 7 |- ( ph -> X e. NN )
17	16	nnrpd 9902	. . . . . 6 |- ( ph -> X e. RR+ )
18		rpcxplogb 15653	. . . . . 6 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B ^c ( B logb X ) ) = X )
19	9, 14, 17, 18	syl3anc 1271	. . . . 5 |- ( ph -> ( B ^c ( B logb X ) ) = X )
20	19	oveq1d 6022	. . . 4 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) = ( X ^ N ) )
21		znq 9831	. . . . . . . 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 9832	. . . . . . 7 |- ( ( M / N ) e. QQ -> ( M / N ) e. RR )
24	22, 23	syl 14	. . . . . 6 |- ( ph -> ( M / N ) e. RR )
25	5	nncnd 9135	. . . . . 6 |- ( ph -> N e. CC )
26	9, 24, 25	cxpmuld 15626	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( ( B ^c ( M / N ) ) ^c N ) )
27	4	zcnd 9581	. . . . . . . 8 |- ( ph -> M e. CC )
28	5	nnap0d 9167	. . . . . . . 8 |- ( ph -> N # 0 )
29	27, 25, 28	divcanap1d 8949	. . . . . . 7 |- ( ph -> ( ( M / N ) x. N ) = M )
30	29	oveq2d 6023	. . . . . 6 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^c M ) )
31		cxpexpnn 15585	. . . . . . 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 2262	. . . . 5 |- ( ph -> ( B ^c ( ( M / N ) x. N ) ) = ( B ^ M ) )
34	9, 24	rpcxpcld 15622	. . . . . 6 |- ( ph -> ( B ^c ( M / N ) ) e. RR+ )
35	5	nnzd 9579	. . . . . 6 |- ( ph -> N e. ZZ )
36		cxpexprp 15584	. . . . . 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 2271	. . . 4 |- ( ph -> ( ( B ^c ( M / N ) ) ^ N ) = ( B ^ M ) )
39	6, 20, 38	3brtr4d 4115	. . 3 |- ( ph -> ( ( B ^c ( B logb X ) ) ^ N ) # ( ( B ^c ( M / N ) ) ^ N ) )
40		relogbzcl 15641	. . . . . . 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 8186	. . . . 5 |- ( ph -> ( B logb X ) e. CC )
43	9, 42	rpcncxpcld 15616	. . . 4 |- ( ph -> ( B ^c ( B logb X ) ) e. CC )
44		qcn 9841	. . . . . 6 |- ( ( M / N ) e. QQ -> ( M / N ) e. CC )
45	22, 44	syl 14	. . . . 5 |- ( ph -> ( M / N ) e. CC )
46	9, 45	rpcncxpcld 15616	. . . 4 |- ( ph -> ( B ^c ( M / N ) ) e. CC )
47		apexp1 10952	. . . 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 1271	. . 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 15628	. . 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 1272	. 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 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   1c1 8011   x. cmul 8015   < clt 8192   # cap 8739   / cdiv 8830   NNcn 9121   2c2 9172   ZZcz 9457   ZZ>=cuz 9733   QQcq 9826   RR+crp 9861   ^cexp 10772   gcd cgcd 12489   ^c ccxp 15546   log_b clogb 15632

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130  ax-pre-suploc 8131  ax-addf 8132  ax-mulf 8133

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-disj 4060  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-of 6224  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-frec 6543  df-1o 6568  df-2o 6569  df-oadd 6572  df-er 6688  df-map 6805  df-pm 6806  df-en 6896  df-dom 6897  df-fin 6898  df-sup 7162  df-inf 7163  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-xneg 9980  df-xadd 9981  df-ioo 10100  df-ico 10102  df-icc 10103  df-fz 10217  df-fzo 10351  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-fac 10960  df-bc 10982  df-ihash 11010  df-shft 11341  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-clim 11805  df-sumdc 11880  df-ef 12174  df-e 12175  df-dvds 12314  df-gcd 12490  df-prm 12645  df-rest 13289  df-topgen 13308  df-psmet 14522  df-xmet 14523  df-met 14524  df-bl 14525  df-mopn 14526  df-top 14687  df-topon 14700  df-bases 14732  df-ntr 14785  df-cn 14877  df-cnp 14878  df-tx 14942  df-cncf 15260  df-limced 15345  df-dvap 15346  df-relog 15547  df-rpcxp 15548  df-logb 15633

This theorem is referenced by:  logbgcd1irrap  15659