MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  logbval Structured version   Visualization version   GIF version

Theorem logbval 26507
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 6890 . . 3 (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
21oveq2d 7427 . 2 (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3 fveq2 6890 . . 3 (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
43oveq1d 7426 . 2 (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5 df-logb 26506 . 2 logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6 ovex 7444 . 2 ((log‘𝑋) / (log‘𝐵)) ∈ V
72, 4, 5, 6ovmpo 7570 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 394   = wceq 1539  wcel 2104  cdif 3944  {csn 4627  {cpr 4629  cfv 6542  (class class class)co 7411  cc 11110  0cc0 11112  1c1 11113   / cdiv 11875  logclog 26299   logb clogb 26505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-logb 26506
This theorem is referenced by:  logbcl  26508  logbid1  26509  logb1  26510  elogb  26511  logbchbase  26512  relogbval  26513  relogbcl  26514  relogbreexp  26516  relogbmul  26518  nnlogbexp  26522  relogbcxp  26526  cxplogb  26527  logbgt0b  26534  dvrelog2b  41237  dvrelogpow2b  41239  rege1logbrege0  47331  logb2aval  47896
  Copyright terms: Public domain W3C validator