Theorem rplogbval 15265

Description: Define the value of the log_b function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by Jim Kingdon, 3-Jul-2024.)

Assertion

Ref	Expression
rplogbval	|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Proof of Theorem rplogbval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpcn 9754	. . . 4 |- ( B e. RR+ -> B e. CC )
2	1	3ad2ant1 1020	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. CC )
3		rpne0 9761	. . . 4 |- ( B e. RR+ -> B =/= 0 )
4	3	3ad2ant1 1020	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 0 )
5		simp2 1000	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B # 1 )
6		1cnd 8059	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> 1 e. CC )
7		apne 8667	. . . . 5 |- ( ( B e. CC /\ 1 e. CC ) -> ( B # 1 -> B =/= 1 ) )
8	2, 6, 7	syl2anc 411	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B # 1 -> B =/= 1 ) )
9	5, 8	mpd 13	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 1 )
10		eldifpr 3650	. . 3 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
11	2, 4, 9, 10	syl3anbrc 1183	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
12		rpcn 9754	. . . 4 |- ( X e. RR+ -> X e. CC )
13	12	3ad2ant3 1022	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. CC )
14		rpne0 9761	. . . 4 |- ( X e. RR+ -> X =/= 0 )
15	14	3ad2ant3 1022	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X =/= 0 )
16		eldifsn 3750	. . 3 |- ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) )
17	13, 15, 16	sylanbrc 417	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. ( CC \ { 0 } ) )
18		simp3 1001	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. RR+ )
19	18	relogcld 15202	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` X ) e. RR )
20		simp1 999	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. RR+ )
21	20	relogcld 15202	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) e. RR )
22	20, 5	logrpap0d 15198	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) # 0 )
23	19, 21, 22	redivclapd 8879	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( ( log ` X ) / ( log ` B ) ) e. RR )
24		fveq2 5561	. . . 4 |- ( x = B -> ( log ` x ) = ( log ` B ) )
25	24	oveq2d 5941	. . 3 |- ( x = B -> ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26		fveq2 5561	. . . 4 |- ( y = X -> ( log ` y ) = ( log ` X ) )
27	26	oveq1d 5940	. . 3 |- ( y = X -> ( ( log ` y ) / ( log ` B ) ) = ( ( log ` X ) / ( log ` B ) ) )
28		df-logb 15264	. . 3 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
29	25, 27, 28	ovmpog 6061	. 2 |- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) /\ ( ( log ` X ) / ( log ` B ) ) e. RR ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
30	11, 17, 23, 29	syl3anc 1249	1 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   \ cdif 3154   {csn 3623   {cpr 3624   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   RRcr 7895   0cc0 7896   1c1 7897   # cap 8625   / cdiv 8716   RR+crp 9745   logclog 15176   log_b clogb 15263

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 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-pre-suploc 8017  ax-addf 8018  ax-mulf 8019

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-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 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-ico 9986  df-icc 9987  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-fac 10835  df-bc 10857  df-ihash 10885  df-shft 10997  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ef 11830  df-e 11831  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891  df-limced 14976  df-dvap 14977  df-relog 15178  df-logb 15264

This theorem is referenced by:  rplogbcl  15266  rplogbid1  15267  rplogb1  15268  rpelogb  15269  rplogbchbase  15270  relogbval  15271  rplogbreexp  15273  rprelogbmul  15275  rpcxplogb  15284  logbgt0b  15286