Theorem rplogbval 14294

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 9660	. . . 4 |- ( B e. RR+ -> B e. CC )
2	1	3ad2ant1 1018	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. CC )
3		rpne0 9667	. . . 4 |- ( B e. RR+ -> B =/= 0 )
4	3	3ad2ant1 1018	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 0 )
5		simp2 998	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B # 1 )
6		1cnd 7972	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> 1 e. CC )
7		apne 8578	. . . . 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 3619	. . 3 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
11	2, 4, 9, 10	syl3anbrc 1181	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
12		rpcn 9660	. . . 4 |- ( X e. RR+ -> X e. CC )
13	12	3ad2ant3 1020	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. CC )
14		rpne0 9667	. . . 4 |- ( X e. RR+ -> X =/= 0 )
15	14	3ad2ant3 1020	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X =/= 0 )
16		eldifsn 3719	. . 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 999	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. RR+ )
19	18	relogcld 14234	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` X ) e. RR )
20		simp1 997	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. RR+ )
21	20	relogcld 14234	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) e. RR )
22	20, 5	logrpap0d 14230	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) # 0 )
23	19, 21, 22	redivclapd 8790	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( ( log ` X ) / ( log ` B ) ) e. RR )
24		fveq2 5515	. . . 4 |- ( x = B -> ( log ` x ) = ( log ` B ) )
25	24	oveq2d 5890	. . 3 |- ( x = B -> ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26		fveq2 5515	. . . 4 |- ( y = X -> ( log ` y ) = ( log ` X ) )
27	26	oveq1d 5889	. . 3 |- ( y = X -> ( ( log ` y ) / ( log ` B ) ) = ( ( log ` X ) / ( log ` B ) ) )
28		df-logb 14293	. . 3 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
29	25, 27, 28	ovmpog 6008	. 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 1238	1 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2148   =/= wne 2347   \ cdif 3126   {csn 3592   {cpr 3593   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811   # cap 8536   / cdiv 8627   RR+crp 9651   logclog 14208   log_b clogb 14292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930  ax-pre-suploc 7931  ax-addf 7932  ax-mulf 7933

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3981  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-of 6082  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-map 6649  df-pm 6650  df-en 6740  df-dom 6741  df-fin 6742  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-xneg 9770  df-xadd 9771  df-ioo 9890  df-ico 9892  df-icc 9893  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-fac 10701  df-bc 10723  df-ihash 10751  df-shft 10819  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-sumdc 11357  df-ef 11651  df-e 11652  df-rest 12684  df-topgen 12703  df-psmet 13378  df-xmet 13379  df-met 13380  df-bl 13381  df-mopn 13382  df-top 13429  df-topon 13442  df-bases 13474  df-ntr 13527  df-cn 13619  df-cnp 13620  df-tx 13684  df-cncf 13989  df-limced 14056  df-dvap 14057  df-relog 14210  df-logb 14293

This theorem is referenced by:  rplogbcl  14295  rplogbid1  14296  rplogb1  14297  rpelogb  14298  rplogbchbase  14299  relogbval  14300  rplogbreexp  14302  rprelogbmul  14304  rpcxplogb  14313  logbgt0b  14315