Theorem rplogbchbase 13278

Description: Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)

Assertion

Ref	Expression
rplogbchbase	|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( A logb X ) = ( ( B logb X ) / ( B logb A ) ) )

Proof of Theorem rplogbchbase

Step	Hyp	Ref	Expression
1		simp3 984	. . . . 5 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> X e. RR+ )
2	1	relogcld 13214	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` X ) e. RR )
3	2	recnd 7906	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` X ) e. CC )
4		simp1l 1006	. . . . 5 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> A e. RR+ )
5	4	relogcld 13214	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` A ) e. RR )
6	5	recnd 7906	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` A ) e. CC )
7		simp2l 1008	. . . . 5 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> B e. RR+ )
8	7	relogcld 13214	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` B ) e. RR )
9	8	recnd 7906	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` B ) e. CC )
10		simp1r 1007	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> A # 1 )
11	4, 10	logrpap0d 13210	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` A ) # 0 )
12		simp2r 1009	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> B # 1 )
13	7, 12	logrpap0d 13210	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( log ` B ) # 0 )
14	3, 6, 9, 11, 13	divcanap7d 8692	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( ( ( log ` X ) / ( log ` B ) ) / ( ( log ` A ) / ( log ` B ) ) ) = ( ( log ` X ) / ( log ` A ) ) )
15		rplogbval 13273	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
16	7, 12, 1, 15	syl3anc 1220	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
17		rplogbval 13273	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ A e. RR+ ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) )
18	7, 12, 4, 17	syl3anc 1220	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) )
19	16, 18	oveq12d 5842	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( ( B logb X ) / ( B logb A ) ) = ( ( ( log ` X ) / ( log ` B ) ) / ( ( log ` A ) / ( log ` B ) ) ) )
20		rplogbval 13273	. . 3 |- ( ( A e. RR+ /\ A # 1 /\ X e. RR+ ) -> ( A logb X ) = ( ( log ` X ) / ( log ` A ) ) )
21	4, 10, 1, 20	syl3anc 1220	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( A logb X ) = ( ( log ` X ) / ( log ` A ) ) )
22	14, 19, 21	3eqtr4rd 2201	1 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( A logb X ) = ( ( B logb X ) / ( B logb A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   1c1 7733   # cap 8456   / cdiv 8545   RR+crp 9560   logclog 13188   log_b clogb 13271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852  ax-pre-suploc 7853  ax-addf 7854  ax-mulf 7855

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-disj 3943  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-of 6032  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-map 6595  df-pm 6596  df-en 6686  df-dom 6687  df-fin 6688  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-xneg 9679  df-xadd 9680  df-ioo 9796  df-ico 9798  df-icc 9799  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-fac 10600  df-bc 10622  df-ihash 10650  df-shft 10715  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251  df-ef 11545  df-e 11546  df-rest 12364  df-topgen 12383  df-psmet 12398  df-xmet 12399  df-met 12400  df-bl 12401  df-mopn 12402  df-top 12407  df-topon 12420  df-bases 12452  df-ntr 12507  df-cn 12599  df-cnp 12600  df-tx 12664  df-cncf 12969  df-limced 13036  df-dvap 13037  df-relog 13190  df-logb 13272

This theorem is referenced by: (None)