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Theorem logbval 25276
 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 David A. Wheeler, 16-Jul-2017.)
Assertion
Ref Expression
logbval ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Proof of Theorem logbval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6669 . . 3 (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
21oveq2d 7166 . 2 (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3 fveq2 6669 . . 3 (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
43oveq1d 7165 . 2 (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5 df-logb 25275 . 2 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6 ovex 7183 . 2 ((log‘𝑋) / (log‘𝐵)) ∈ V
72, 4, 5, 6ovmpo 7304 1 ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ∖ cdif 3937  {csn 4564  {cpr 4566  ‘cfv 6354  (class class class)co 7150  ℂcc 10529  0cc0 10531  1c1 10532   / cdiv 11291  logclog 25070   logb clogb 25274 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-logb 25275 This theorem is referenced by:  logbcl  25277  logbid1  25278  logb1  25279  elogb  25280  logbchbase  25281  relogbval  25282  relogbcl  25283  relogbreexp  25285  relogbmul  25287  nnlogbexp  25291  relogbcxp  25295  cxplogb  25296  logbgt0b  25303  rege1logbrege0  44520  logb2aval  44765
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