Theorem logbval 26827

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 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.

Step	Hyp	Ref	Expression
1		fveq2 6920	. . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
2	1	oveq2d 7464	. 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3		fveq2 6920	. . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
4	3	oveq1d 7463	. 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5		df-logb 26826	. 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6		ovex 7481	. 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V
7	2, 4, 5, 6	ovmpo 7610	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  {csn 4648  {cpr 4650  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  0cc0 11184  1c1 11185   / cdiv 11947  logclog 26614   log_b clogb 26825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-logb 26826

This theorem is referenced by:  logbcl  26828  logbid1  26829  logb1  26830  elogb  26831  logbchbase  26832  relogbval  26833  relogbcl  26834  relogbreexp  26836  relogbmul  26838  nnlogbexp  26842  relogbcxp  26846  cxplogb  26847  logbgt0b  26854  dvrelog2b  42023  dvrelogpow2b  42025  rege1logbrege0  48292  logb2aval  48856