Theorem logbval 24703

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 6352	. . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
2	1	oveq2d 6829	. 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3		fveq2 6352	. . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
4	3	oveq1d 6828	. 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5		df-logb 24702	. 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6		ovex 6841	. 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V
7	2, 4, 5, 6	ovmpt2 6961	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ∖ cdif 3712  {csn 4321  {cpr 4323  ‘cfv 6049  (class class class)co 6813  ℂcc 10126  0cc0 10128  1c1 10129   / cdiv 10876  logclog 24500   log_b clogb 24701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-logb 24702

This theorem is referenced by:  logbcl  24704  logbid1  24705  logb1  24706  elogb  24707  logbchbase  24708  relogbval  24709  relogbcl  24710  relogbreexp  24712  relogbmul  24714  nnlogbexp  24718  relogbcxp  24722  cxplogb  24723  rege1logbrege0  42862  logb2aval  43018