Theorem rplogbval 15492

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 9804	. . . 4 |- ( B e. RR+ -> B e. CC )
2	1	3ad2ant1 1021	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. CC )
3		rpne0 9811	. . . 4 |- ( B e. RR+ -> B =/= 0 )
4	3	3ad2ant1 1021	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B =/= 0 )
5		simp2 1001	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B # 1 )
6		1cnd 8108	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> 1 e. CC )
7		apne 8716	. . . . 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 3665	. . 3 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
11	2, 4, 9, 10	syl3anbrc 1184	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) )
12		rpcn 9804	. . . 4 |- ( X e. RR+ -> X e. CC )
13	12	3ad2ant3 1023	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. CC )
14		rpne0 9811	. . . 4 |- ( X e. RR+ -> X =/= 0 )
15	14	3ad2ant3 1023	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X =/= 0 )
16		eldifsn 3766	. . 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 1002	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> X e. RR+ )
19	18	relogcld 15429	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` X ) e. RR )
20		simp1 1000	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> B e. RR+ )
21	20	relogcld 15429	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) e. RR )
22	20, 5	logrpap0d 15425	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( log ` B ) # 0 )
23	19, 21, 22	redivclapd 8928	. 2 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( ( log ` X ) / ( log ` B ) ) e. RR )
24		fveq2 5589	. . . 4 |- ( x = B -> ( log ` x ) = ( log ` B ) )
25	24	oveq2d 5973	. . 3 |- ( x = B -> ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26		fveq2 5589	. . . 4 |- ( y = X -> ( log ` y ) = ( log ` X ) )
27	26	oveq1d 5972	. . 3 |- ( y = X -> ( ( log ` y ) / ( log ` B ) ) = ( ( log ` X ) / ( log ` B ) ) )
28		df-logb 15491	. . 3 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
29	25, 27, 28	ovmpog 6093	. 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 1250	1 |- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2177   =/= wne 2377   \ cdif 3167   {csn 3638   {cpr 3639   class class class wbr 4051   ` cfv 5280  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   1c1 7946   # cap 8674   / cdiv 8765   RR+crp 9795   logclog 15403   log_b clogb 15490

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065  ax-pre-suploc 8066  ax-addf 8067  ax-mulf 8068

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-disj 4028  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-of 6171  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-frec 6490  df-1o 6515  df-oadd 6519  df-er 6633  df-map 6750  df-pm 6751  df-en 6841  df-dom 6842  df-fin 6843  df-sup 7101  df-inf 7102  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-xneg 9914  df-xadd 9915  df-ioo 10034  df-ico 10036  df-icc 10037  df-fz 10151  df-fzo 10285  df-seqfrec 10615  df-exp 10706  df-fac 10893  df-bc 10915  df-ihash 10943  df-shft 11201  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-clim 11665  df-sumdc 11740  df-ef 12034  df-e 12035  df-rest 13148  df-topgen 13167  df-psmet 14380  df-xmet 14381  df-met 14382  df-bl 14383  df-mopn 14384  df-top 14545  df-topon 14558  df-bases 14590  df-ntr 14643  df-cn 14735  df-cnp 14736  df-tx 14800  df-cncf 15118  df-limced 15203  df-dvap 15204  df-relog 15405  df-logb 15491

This theorem is referenced by:  rplogbcl  15493  rplogbid1  15494  rplogb1  15495  rpelogb  15496  rplogbchbase  15497  relogbval  15498  rplogbreexp  15500  rprelogbmul  15502  rpcxplogb  15511  logbgt0b  15513