Theorem rplogbval 15936

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 10013	. . . 4 |- ( B e. RR+ -> B e. CC )
2	1	3ad2ant1 1045	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. CC )
3		rpne0 10020	. . . 4 |- ( B e. RR+ -> B =/= 0 )
4	3	3ad2ant1 1045	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 0 )
5		simp2 1025	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B # 1 )
6		1cnd 8306	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> 1 e. CC )
7		apne 8914	. . . . 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 3721	. . 3 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
11	2, 4, 9, 10	syl3anbrc 1208	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
12		rpcn 10013	. . . 4 |- ( X e. RR+ -> X e. CC )
13	12	3ad2ant3 1047	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. CC )
14		rpne0 10020	. . . 4 |- ( X e. RR+ -> X =/= 0 )
15	14	3ad2ant3 1047	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X =/= 0 )
16		eldifsn 3825	. . 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 1026	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. RR+ )
19	18	relogcld 15873	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` X ) e. RR )
20		simp1 1024	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. RR+ )
21	20	relogcld 15873	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) e. RR )
22	20, 5	logrpap0d 15869	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) # 0 )
23	19, 21, 22	redivclapd 9126	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( ( log ` X ) / ( log ` B ) ) e. RR )
24		fveq2 5675	. . . 4 |- ( x = B -> ( log ` x ) = ( log ` B ) )
25	24	oveq2d 6074	. . 3 |- ( x = B -> ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26		fveq2 5675	. . . 4 |- ( y = X -> ( log ` y ) = ( log ` X ) )
27	26	oveq1d 6073	. . 3 |- ( y = X -> ( ( log ` y ) / ( log ` B ) ) = ( ( log ` X ) / ( log ` B ) ) )
28		df-logb 15935	. . 3 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
29	25, 27, 28	ovmpog 6196	. 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 1274	1 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   \ cdif 3211   {csn 3694   {cpr 3695   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   # cap 8872   / cdiv 8963   RR+crp 10004   logclog 15847   log_b clogb 15934

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263  ax-pre-suploc 8264  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-disj 4091  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-of 6275  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-map 6897  df-pm 6898  df-en 6989  df-dom 6990  df-fin 6991  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-xneg 10124  df-xadd 10125  df-ioo 10244  df-ico 10246  df-icc 10247  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-fac 11113  df-bc 11135  df-ihash 11164  df-shft 11525  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064  df-ef 12359  df-e 12360  df-rest 13538  df-topgen 13557  df-psmet 14817  df-xmet 14818  df-met 14819  df-bl 14820  df-mopn 14821  df-top 14989  df-topon 15002  df-bases 15034  df-ntr 15087  df-cn 15179  df-cnp 15180  df-tx 15244  df-cncf 15562  df-limced 15647  df-dvap 15648  df-relog 15849  df-logb 15935

This theorem is referenced by:  rplogbcl  15937  rplogbid1  15938  rplogb1  15939  rpelogb  15940  rplogbchbase  15941  relogbval  15942  rplogbreexp  15944  rprelogbmul  15946  rpcxplogb  15955  logbgt0b  15957