Theorem rplogbval 15613

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 9854	. . . 4 |- ( B e. RR+ -> B e. CC )
2	1	3ad2ant1 1042	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. CC )
3		rpne0 9861	. . . 4 |- ( B e. RR+ -> B =/= 0 )
4	3	3ad2ant1 1042	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 0 )
5		simp2 1022	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B # 1 )
6		1cnd 8158	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> 1 e. CC )
7		apne 8766	. . . . 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 3693	. . 3 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
11	2, 4, 9, 10	syl3anbrc 1205	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
12		rpcn 9854	. . . 4 |- ( X e. RR+ -> X e. CC )
13	12	3ad2ant3 1044	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. CC )
14		rpne0 9861	. . . 4 |- ( X e. RR+ -> X =/= 0 )
15	14	3ad2ant3 1044	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X =/= 0 )
16		eldifsn 3794	. . 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 1023	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. RR+ )
19	18	relogcld 15550	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` X ) e. RR )
20		simp1 1021	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. RR+ )
21	20	relogcld 15550	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) e. RR )
22	20, 5	logrpap0d 15546	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) # 0 )
23	19, 21, 22	redivclapd 8978	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( ( log ` X ) / ( log ` B ) ) e. RR )
24		fveq2 5626	. . . 4 |- ( x = B -> ( log ` x ) = ( log ` B ) )
25	24	oveq2d 6016	. . 3 |- ( x = B -> ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26		fveq2 5626	. . . 4 |- ( y = X -> ( log ` y ) = ( log ` X ) )
27	26	oveq1d 6015	. . 3 |- ( y = X -> ( ( log ` y ) / ( log ` B ) ) = ( ( log ` X ) / ( log ` B ) ) )
28		df-logb 15612	. . 3 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
29	25, 27, 28	ovmpog 6138	. 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 1271	1 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   \ cdif 3194   {csn 3666   {cpr 3667   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   1c1 7996   # cap 8724   / cdiv 8815   RR+crp 9845   logclog 15524   log_b clogb 15611

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115  ax-pre-suploc 8116  ax-addf 8117  ax-mulf 8118

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 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-of 6216  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-map 6795  df-pm 6796  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-xneg 9964  df-xadd 9965  df-ioo 10084  df-ico 10086  df-icc 10087  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-bc 10965  df-ihash 10993  df-shft 11321  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860  df-ef 12154  df-e 12155  df-rest 13269  df-topgen 13288  df-psmet 14501  df-xmet 14502  df-met 14503  df-bl 14504  df-mopn 14505  df-top 14666  df-topon 14679  df-bases 14711  df-ntr 14764  df-cn 14856  df-cnp 14857  df-tx 14921  df-cncf 15239  df-limced 15324  df-dvap 15325  df-relog 15526  df-logb 15612

This theorem is referenced by:  rplogbcl  15614  rplogbid1  15615  rplogb1  15616  rpelogb  15617  rplogbchbase  15618  relogbval  15619  rplogbreexp  15621  rprelogbmul  15623  rpcxplogb  15632  logbgt0b  15634