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Theorem rplogbval 14030
Description: Define the value of the logb 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.
StepHypRef Expression
1 rpcn 9649 . . . 4  |-  ( B  e.  RR+  ->  B  e.  CC )
213ad2ant1 1018 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  B  e.  CC )
3 rpne0 9656 . . . 4  |-  ( B  e.  RR+  ->  B  =/=  0 )
433ad2ant1 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 7964 . . . . 5  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  1  e.  CC )
7 apne 8570 . . . . 5  |-  ( ( B  e.  CC  /\  1  e.  CC )  ->  ( B #  1  ->  B  =/=  1 ) )
82, 6, 7syl2anc 411 . . . 4  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( B #  1  ->  B  =/=  1
) )
95, 8mpd 13 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  B  =/=  1
)
10 eldifpr 3618 . . 3  |-  ( B  e.  ( CC  \  { 0 ,  1 } )  <->  ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 ) )
112, 4, 9, 10syl3anbrc 1181 . 2  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  B  e.  ( CC  \  { 0 ,  1 } ) )
12 rpcn 9649 . . . 4  |-  ( X  e.  RR+  ->  X  e.  CC )
13123ad2ant3 1020 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  X  e.  CC )
14 rpne0 9656 . . . 4  |-  ( X  e.  RR+  ->  X  =/=  0 )
15143ad2ant3 1020 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  X  =/=  0
)
16 eldifsn 3718 . . 3  |-  ( X  e.  ( CC  \  { 0 } )  <-> 
( X  e.  CC  /\  X  =/=  0 ) )
1713, 15, 16sylanbrc 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+ )
1918relogcld 13970 . . 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+ )
2120relogcld 13970 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( log `  B
)  e.  RR )
2220, 5logrpap0d 13966 . . 3  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( log `  B
) #  0 )
2319, 21, 22redivclapd 8781 . 2  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( ( log `  X )  /  ( log `  B ) )  e.  RR )
24 fveq2 5511 . . . 4  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
2524oveq2d 5885 . . 3  |-  ( x  =  B  ->  (
( log `  y
)  /  ( log `  x ) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
26 fveq2 5511 . . . 4  |-  ( y  =  X  ->  ( log `  y )  =  ( log `  X
) )
2726oveq1d 5884 . . 3  |-  ( y  =  X  ->  (
( log `  y
)  /  ( log `  B ) )  =  ( ( log `  X
)  /  ( log `  B ) ) )
28 df-logb 14029 . . 3  |- logb  =  (
x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
2925, 27, 28ovmpog 6003 . 2  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } )  /\  ( ( log `  X )  /  ( log `  B ) )  e.  RR )  -> 
( B logb  X )  =  ( ( log `  X
)  /  ( log `  B ) ) )
3011, 17, 23, 29syl3anc 1238 1  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( B logb  X )  =  ( ( log `  X )  /  ( log `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347    \ cdif 3126   {csn 3591   {cpr 3592   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   # cap 8528    / cdiv 8618   RR+crp 9640   logclog 13944   logb clogb 14028
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922  ax-pre-suploc 7923  ax-addf 7924  ax-mulf 7925
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-of 6077  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-map 6644  df-pm 6645  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-ioo 9879  df-ico 9881  df-icc 9882  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-shft 10808  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640  df-e 11641  df-rest 12638  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-met 13156  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cn 13355  df-cnp 13356  df-tx 13420  df-cncf 13725  df-limced 13792  df-dvap 13793  df-relog 13946  df-logb 14029
This theorem is referenced by:  rplogbcl  14031  rplogbid1  14032  rplogb1  14033  rpelogb  14034  rplogbchbase  14035  relogbval  14036  rplogbreexp  14038  rprelogbmul  14040  rpcxplogb  14049  logbgt0b  14051
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