Theorem logbval 26676

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 6858	. . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
2	1	oveq2d 7403	. 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3		fveq2 6858	. . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
4	3	oveq1d 7402	. 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5		df-logb 26675	. 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6		ovex 7420	. 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V
7	2, 4, 5, 6	ovmpo 7549	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3911  {csn 4589  {cpr 4591  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  0cc0 11068  1c1 11069   / cdiv 11835  logclog 26463   log_b clogb 26674

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-logb 26675

This theorem is referenced by:  logbcl  26677  logbid1  26678  logb1  26679  elogb  26680  logbchbase  26681  relogbval  26682  relogbcl  26683  relogbreexp  26685  relogbmul  26687  nnlogbexp  26691  relogbcxp  26695  cxplogb  26696  logbgt0b  26703  dvrelog2b  42054  dvrelogpow2b  42056  rege1logbrege0  48547  logb2aval  49753